#625839

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2672 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)(10,14)(11,16)(17,18,23,26)(19,21,20,24)(22,30,31,25)(27,32,28,29);(6,8)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,32,30,29);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,32,30,29);(1,2,3,4)(9,11,13,16)(10,15,14,12)(17,21,23,24)(18,19,26,20)(22,29,31,32)(25,27,30,28);(9,12,13,15)(10,16,14,11)(17,22,23,31)(18,25,26,30)(19,27,20,28)(21,29,24,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=i,e^2=fg,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, 533035);