#675001

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

2090 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)(5,6,9,8)(7,10,12,11)(13,14)(15,17)(16,18)(19,20)(23,28)(24,26);(6,8)(10,11)(13,14,16,18)(15,19,20,17)(21,22,25,27)(23,26,28,24);(5,7,9,12)(6,10,8,11)(13,14)(15,17)(16,18)(19,20)(21,23)(22,26)(24,27)(25,28);(1,2,3,4)(21,24,25,26)(22,23,27,28);(5,8,9,6)(7,11,12,10)(13,15,16,20)(14,17,18,19)

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=fh,d^2=i,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, 582197);