#900804

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2286 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,e^2=fi,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, 808000);