#687595

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#548	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6618 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)(13,14,19,16)(15,21,17,20)(18,22,26,24)(23,28,25,27);(10,12)(13,15,19,17)(14,20,16,21)(18,25,26,23)(22,28,24,27);(5,6)(7,8)(9,10,11,12)(13,16,19,14)(15,20,17,21)(18,24,26,22)(23,27,25,28);(1,2)(3,4)(5,6)(7,8)(13,17,19,15)(14,21,16,20)(18,25,26,23)(22,28,24,27);(13,18,19,26)(14,22,16,24)(15,23,17,25)(20,27,21,28)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=f,e^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, 594791);