#616892

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2297 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)(9,10,13,16)(11,15,12,14)(18,22)(19,20)(25,26)(27,29)(28,31)(30,32);(9,11,13,12)(10,14,16,15)(17,18)(19,23)(20,24)(21,22)(26,27)(30,31);(17,19,21,20)(18,23,22,24)(25,27,29,26)(28,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,20,21,19)(18,24,22,23);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,28,29,32)(26,30,27,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,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, 524088);