#1462965

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1950 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=hi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

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, 1370161);