#154157

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2978 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61353);