#840684

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

3096 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)(6,9)(10,12)(14,15)(18,19)(21,22)(23,25)(24,27)(26,28);(13,14,17,15)(16,19,20,18)(21,22,24,27)(23,26,28,25);(1,2)(3,4)(13,14,17,15)(16,18,20,19)(21,23)(22,25)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,18,20,19)(21,23)(22,26)(24,28)(25,27);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,18,15,19)(21,22)(23,25)(24,27)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 747880);