#6357948

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357939
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1744 conjugacy classes of subgroups; only the 566 normal subgroups are shown.

normal subgroups only

This group has 1744 conjugacy classes of subgroups — too many to draw. Only its 566 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,7,11)(3,12,14,27)(4,9,17,23)(5,10,6,20)(8,15,22,16)(13,26,29,31)(18,24,19,25)(21,30,32,28)(33,34,38,42)(35,39,44,52)(36,46,47,61)(37,49,50,41)(40,54,55,48)(43,56,57,64)(45,60,59,53)(51,62,63,58);(1,3,5,15,7,14,6,16)(2,8,20,27,11,22,10,12)(4,13,19,30,17,29,18,28)(9,21,24,31,23,32,25,26)(33,35)(34,39)(36,43)(37,45)(38,44)(40,51)(41,53)(42,52)(46,56)(47,57)(48,58)(49,60)(50,59)(54,62)(55,63)(61,64);(1,4)(2,9)(3,13)(5,19)(6,18)(7,17)(8,21)(10,25)(11,23)(12,26)(14,29)(15,30)(16,28)(20,24)(22,32)(27,31)(33,36,37,48,38,47,50,54)(34,40,41,46,42,55,49,61)(35,43,45,58,44,57,59,62)(39,51,53,56,52,63,60,64);(3,14)(4,17)(8,22)(9,23)(12,27)(15,16)(18,19)(24,25)(36,47)(40,55)(43,57)(46,61)(48,54)(51,63)(56,64)(58,62);(1,5,7,6)(2,10,11,20)(3,15,14,16)(4,18,17,19)(8,12,22,27)(9,24,23,25)(13,28,29,30)(21,31,32,26)(35,44)(39,52)(43,57)(45,59)(51,63)(53,60)(56,64)(58,62)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,e^b=ei,d^c=dhi,e^c=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265144);