#7104773

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1858 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)(21,22,28,36,27,31,26,25)(23,30,35,29,24,34,33,32);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,29)(24,27)(25,34)(26,35)(28,33)(30,36)(31,32);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,27,23)(22,30,31,34)(25,32,36,29)(26,35,28,33);(8,16)(11,20)(13,15)(17,19)(21,25,28,22,27,36,26,31)(23,32,35,30,24,29,33,34);(6,12)(7,14)(11,20)(13,15)(22,31)(23,24)(25,36)(33,35)

Polycyclic presentation:

2^9|a^2=fh,c^2=h,d^2=fi,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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250887, #72257115, #72257127, …
maximal normal subgroup of	#72250887, #72257115, #72257127, …
maximal quotient of	#72264659

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7011969);