#298596

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
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×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=fh,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,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, 205792);