#4310243

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,3)(5,6,8,10)(7,11,12,9)(13,14,17,19)(15,20,16,18)(21,22)(23,24)(25,33)(26,27)(28,29)(30,34)(31,35)(32,36);(5,6)(7,9)(8,10)(11,12)(13,15,17,16)(14,18,19,20)(22,27)(23,29)(30,33)(31,36);(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,16,17,15)(14,18,19,20)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,33)(34,36);(1,3,4,2)(5,7,8,12)(6,9,10,11)(13,14)(15,18)(16,20)(17,19)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31);(21,25,26,34)(22,30,27,33)(23,31,29,36)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=fh,b^2=gi,c^2=fg,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4217439);