#763820

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

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

5742 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=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, 671016);