#297456

order	512
nilpotency class	3
isoclinism	isoclinic class of #272795
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	C₂×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1188 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)(5,6,10,14)(7,11,16,19)(8,17,15,20)(9,13,18,12)(21,22)(23,26)(24,27)(25,28)(29,36)(30,34)(31,33)(32,35);(1,2,4,3)(5,7)(6,11)(8,18)(9,15)(10,16)(12,17)(13,20)(14,19)(21,23)(22,26)(24,31)(25,30)(27,33)(28,34)(29,32)(35,36);(5,8,9,7,10,15,18,16)(6,11,12,17,14,19,13,20)(21,24)(22,27)(23,29,30,31)(25,32)(26,33,34,36)(28,35);(1,3,4,2)(24,32)(27,35)(29,31)(33,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=e,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,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204652);