#234449

order	512
nilpotency class	3
isoclinism	isoclinic class of #234438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1157 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1157 conjugacy classes of subgroups — too many to draw. Only its 190 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,d^c=dhi,e^c=eh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141645);