#778777

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2338 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)(13,14,18,17)(15,16,20,19)(21,22,25,24)(23,27,28,26);(5,7)(6,11)(8,12)(9,10)(13,15,18,20)(14,19,17,16);(5,6,10,8)(7,12,9,11)(13,14,18,17)(15,19,20,16)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(5,8,10,6)(7,12,9,11)(13,16)(14,15)(17,20)(18,19)(21,24,25,22)(23,27,28,26);(5,9,10,7)(6,11,8,12)(13,17,18,14)(15,19,20,16)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 685973);