#679795

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2462 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fhi,d^2=i,e^2=fg,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, 586991);