#633949

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1942 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,d^2=f,e^2=hi,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, 541145);