#874354

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2238 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)(7,11)(8,10)(9,12)(13,14,17,20)(15,19,16,18)(21,22,25,27)(23,26,24,28);(5,6,8,10)(7,9,12,11)(13,15,17,16)(14,18,20,19)(21,23,25,24)(22,26,27,28);(5,6)(7,9)(8,10)(11,12)(21,24,25,23)(22,26,27,28);(1,2,3,4)(21,22)(23,28)(24,26)(25,27);(5,7,8,12)(6,9,10,11)(13,16,17,15)(14,19,20,18)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=g,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 781550);