#678068

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1952 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=fg,d^2=ghi,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, 585264);