#4464161

order	512
nilpotency class	2
isoclinism	isoclinic class of #4462490
automorphism	Aut(G) order 12582912
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2586 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4371357);