#4462656

order	512
nilpotency class	2
isoclinism	isoclinic class of #4462490
automorphism	Aut(G) order 4194304
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2568 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)(5,6)(7,10)(8,9)(11,12)(14,15)(16,20)(22,26)(23,29)(24,31)(34,36);(6,8)(10,12)(13,14,17,15)(16,18,20,19)(22,26)(23,29)(28,33)(30,35);(5,7,9,11)(6,10,8,12)(13,15,17,14)(16,18,20,19)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(1,2)(3,4)(5,8,9,6)(7,10,11,12)(13,16,17,20)(14,18,15,19)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36);(21,24)(22,28)(23,30)(25,31)(26,33)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|b^2=gh,c^2=fh,d^2=f,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, 4369852);