#156409

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2610 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egi,c^2=fg,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 63605);