#503272

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3417	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,4)(5,6,11,18)(7,20,24,50)(8,26,30,15)(9,32,34,49)(10,37,38,67)(12,19,13,42)(14,43,46,23)(16,22,53,33)(17,39,55,21)(25,57,40,58)(27,63,61,54)(28,47,64,48)(29,56,60,66)(31,52,41,45)(35,44,36,65)(51,59,62,68);(5,7,21,28,13,25,38,43,11,24,39,64,12,40,10,23)(6,14,37,58,19,48,55,50,18,46,67,57,42,47,17,20)(8,27,59,36,31,22,60,34,30,61,68,35,41,33,29,9)(15,49,66,53,52,65,62,63,26,32,56,16,45,44,51,54);(1,2,3,4)(5,8,11,30)(6,15,18,26)(7,22,24,33)(9,28,34,64)(10,29,38,60)(12,41,13,31)(14,44,46,65)(16,50,53,20)(17,51,55,62)(19,52,42,45)(21,59,39,68)(23,36,43,35)(25,61,40,27)(32,48,49,47)(37,66,67,56)(54,57,63,58);(5,9,21,27,13,36,38,22,11,34,39,61,12,35,10,33)(6,16,37,44,19,54,55,49,18,53,67,65,42,63,17,32)(7,8,28,59,25,31,43,60,24,30,64,68,40,41,23,29)(14,45,58,51,48,15,50,66,46,52,57,62,47,26,20,56)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410468);