#647241

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2480 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=g,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 554437);