#355321

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1218 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)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22,25,28,24,26,23,27);(1,2,3,4)(6,9)(7,11)(22,26)(27,28);(5,7,8,11)(6,10,9,12)(14,17)(18,20)(21,23,24,25)(22,27,26,28);(6,9)(10,12)(13,15,16,19)(14,18,17,20)(22,26)(27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cei

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, 262517);