#7042577

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,25,30,26,31,29,35)(23,32,33,36,24,28,34,27);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,27)(24,26)(25,34)(28,30)(29,33)(31,36)(32,35);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,26,23)(22,28,31,32)(25,34,29,33)(27,35,36,30);(8,16)(11,20)(13,15)(17,19)(21,22,29,35,26,31,25,30)(23,28,34,36,24,32,33,27);(6,12)(7,14)(11,20)(13,15)(23,24)(27,36)(28,32)(33,34)

Polycyclic presentation:

2^9|a^2=fhi,c^2=gi,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308416, #72308641, #72308732
maximal normal subgroup of	#72308416, #72308641, #72308732

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6949773);