#687272

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2430	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5264 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)(6,8)(9,10,11,12)(13,14)(15,19)(16,23)(17,18)(20,25)(21,28)(22,26)(24,27);(9,10,11,12)(14,18)(19,26)(20,23)(27,28);(5,6)(7,8)(9,10)(11,12)(13,15,17,22)(14,19,18,26)(16,24,25,21)(20,28,23,27);(1,2)(3,4)(5,6)(7,8)(13,16,17,25)(14,20,18,23)(15,21,22,24)(19,27,26,28);(13,14)(15,19)(16,20)(17,18)(21,27)(22,26)(23,25)(24,28)

Polycyclic presentation:

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

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, 594468);