#674125

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

1878 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgi,c^2=gi,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 581321);