#365871

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂³×C₈
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=ef,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273067);