#261482

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#39412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1277 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=h,d^2=ei,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=di,f^c=fh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 168678);