#879160

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2588 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=f,d^2=fi,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, 786356);