#726174

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1768 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,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, 633370);