#659635

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

2198 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)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(23,27)(25,28);(2,4)(5,7,11,9)(6,8,12,10)(14,17)(15,19)(21,22,24,26)(23,25,27,28);(1,2,3,4)(5,8,11,10)(6,9,12,7)(13,14)(15,18)(16,17)(19,20);(5,9,11,7)(6,8,12,10)(21,23)(22,25)(24,27)(26,28);(5,10,11,8)(6,7,12,9)(13,15,16,19)(14,18,17,20)

Polycyclic presentation:

2^9|a^2=fgh,b^2=ghi,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, 566831);