#651748

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

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, 558944);