#616668

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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

2390 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)(6,8)(7,11)(14,18)(16,19)(21,22,25,23)(24,26,28,27);(5,6,9,8)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,23,25,22)(24,26,28,27);(1,2)(3,4)(21,24,25,28)(22,26,23,27);(5,7)(6,10)(8,12)(9,11)(13,15,17,20)(14,19,18,16);(5,8,9,6)(7,12,11,10)(13,16,17,19)(14,20,18,15)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 523864);