#875300

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1958 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fg,d^2=fhi,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, 782496);