#686199

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2706 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=fi,e^2=fi,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, 593395);