#686270

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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

(2,4)(10,12)(13,14)(15,16)(17,20)(18,19);(6,8)(10,12)(14,19)(15,17);(5,6)(7,8)(9,10,11,12)(13,15,18,17)(14,20,19,16);(1,2)(3,4)(13,16,18,20)(14,15,19,17);(13,17,18,15)(14,20,19,16)

Polycyclic presentation:

2^9|c^2=g,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.

Relationships

maximal subgroup of	#72297609, #72297614, #72297628
maximal normal subgroup of	#72297609, #72297614, #72297628
maximal quotient of	#72298335, #72298416

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 593466);