#297914

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1056 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)(5,6,8,10)(7,13,15,20)(9,17,14,12)(11,19,18,16)(21,22,27,32)(23,28,34,29)(24,35,33,36)(25,31,26,30);(1,2,4,3)(6,10)(11,18)(12,17)(13,20)(21,23,27,34)(22,28,32,29)(24,25,33,26)(30,35,31,36);(5,7,14,19,8,15,9,16)(6,11,17,20,10,18,12,13)(21,24,26,23,27,33,25,34)(22,29,31,36,32,28,30,35);(1,3,4,2)(6,10)(7,15)(12,17)(16,19)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fh,d^2=ehi,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, 205110);