#624977

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1734 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=fi,d^2=h,e^2=g,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, 532173);