#674134

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

1939 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=fgi,d^2=gh,e^2=fgh,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, 581330);