#651497

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=fg,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, 558693);