#678062

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=fg,d^2=fghi,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, 585258);