#845229

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2070 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=h,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752425);