#297917

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
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	C₂×C₄²	C₂⁴
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205113);