#6677726

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#31021	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1785 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)(21,22,25,28)(23,31,34,36)(24,30,26,29)(27,35,32,33);(1,2)(3,4)(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,22)(23,32)(24,29)(25,28)(26,30)(27,34)(31,35)(33,36);(2,4)(21,23)(22,27)(24,33)(25,34)(26,35)(28,32)(29,31)(30,36);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(22,28)(27,32)(29,30)(31,36);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,25,26)(22,29,28,30)(23,33,34,35)(27,31,32,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=fi,e^2=f,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

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, 6584922);