#298124

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
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

1096 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)(5,6,9,12)(7,14,16,11)(8,17,19,20)(10,15,18,13)(21,22,26,30)(23,31,34,35)(24,28,25,29)(27,32,36,33);(1,2,4,3)(5,7,9,16)(6,11,12,14)(8,13,19,15)(10,17,18,20)(23,32)(24,25)(27,35)(28,29)(31,36)(33,34);(5,8,18,14,9,19,10,11)(6,7,15,20,12,16,13,17)(21,23,25,32,26,34,24,33)(22,27,28,35,30,36,29,31);(1,3,4,2)(6,12)(7,16)(13,15)(17,20)(21,24,26,25)(22,28,30,29)(23,32,34,33)(27,31,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205320);