#6812041

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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
prime radical	#35570	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1741 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6719237);