#616911

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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

2230 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)(6,8)(10,11)(12,16)(17,18,21,24)(19,23,20,22)(25,26)(27,31)(28,29)(30,32);(9,10,13,11)(12,14,16,15)(17,19,21,20)(18,22,24,23)(26,29)(30,31);(17,19,21,20)(18,22,24,23)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,20,21,19)(18,23,24,22);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27,28,32)(26,30,29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=g,d^2=fh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 524107);