#874108

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2537	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3084 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)(5,6)(7,12)(8,10)(9,11)(13,14,17,15)(16,19,20,18)(22,26)(24,27);(6,8)(11,12)(14,15)(18,19)(21,22)(23,27)(24,28)(25,26);(5,7,10,9)(6,11,8,12)(13,15,17,14)(16,19,20,18)(21,23,25,28)(22,27,26,24);(1,2,3,4)(5,8,10,6)(7,12,9,11);(5,9,10,7)(6,12,8,11)(13,16,17,20)(14,18,15,19)(21,24,25,27)(22,28,26,23)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=i,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, 781304);