#4435443

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=gh,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

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, 4342639);