#684066

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

1882 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,23,24,20)(26,27)(30,31);(2,6)(4,7)(18,22)(19,24)(25,26,29,27)(28,31,32,30);(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,22)(21,24)(25,27,29,26)(28,31,32,30);(1,3,5,8)(2,7,6,4)(9,10)(11,13)(12,14)(15,16)(17,20,21,23)(18,19,22,24);(1,4,5,7)(2,8,6,3)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,22)(21,24)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=fgi,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, 591262);