#7104710

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308223, #72308561
maximal normal subgroup of	#72308223, #72308561

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7011906);