#245143

order	512
nilpotency class	3
isoclinism	isoclinic class of #245126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1021 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1021 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,7)(4,8)(5,9)(6,10)(11,18)(12,19)(13,20)(14,28)(15,23)(16,24)(17,25)(21,27)(22,30)(26,32)(29,31)(33,34,38,42)(35,39,44,53)(36,46,48,56)(37,49,51,41)(40,50,55,47)(43,57,58,64)(45,61,60,54)(52,62,63,59);(1,3,6,13)(2,7,10,20)(4,11,15,27)(5,16,17,12)(8,21,23,18)(9,24,25,19)(14,29,30,26)(22,31,28,32)(33,35,38,44)(34,39,42,53)(36,43,48,58)(37,45,51,60)(40,52,55,63)(41,54,49,61)(46,57,56,64)(47,59,50,62);(1,4)(2,8)(3,11)(5,14)(6,15)(7,18)(9,22)(10,23)(12,26)(13,27)(16,29)(17,30)(19,31)(20,21)(24,32)(25,28)(33,36,38,48)(34,40,42,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(41,56,49,46)(45,62,60,59)(54,64,61,57);(1,5,6,17)(2,9,10,25)(3,12,13,16)(4,14,15,30)(7,19,20,24)(8,22,23,28)(11,26,27,29)(18,31,21,32)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=dhi,e^c=eh,f^c=fi,f^d=fh

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, 152339);