#617200

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

3250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(6,8)(9,10,12,14)(11,15,16,13)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31);(6,8)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31);(5,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,20,22,18)(19,23,27,24)(21,28,29,25)(26,30,32,31);(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 524396);