#688485

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2384 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=fg,e^2=f,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, 595681);