#684068

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

2290 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,24,21,20)(25,26,29,28)(27,31,32,30);(1,3,6,4)(2,7,8,5)(17,19,22,21)(18,20,23,24)(26,28)(27,32);(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,23)(22,24)(25,27,29,32)(26,30,28,31);(1,4,6,3)(2,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,21,22,19)(18,20,23,24);(1,5,6,7)(2,3,8,4)(9,11,12,15)(10,13,14,16)(17,20)(18,19)(21,23)(22,24)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,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, 591264);