#4395440

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1950 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)(7,11)(8,10)(9,12)(15,19)(17,20)(21,22,26,23)(24,29,31,27)(25,30,33,28)(32,34,36,35);(5,7,8,12)(6,9,10,11)(13,14,16,18)(15,17,19,20)(22,23)(24,31)(28,30)(32,36);(1,2,4,3)(5,6,8,10)(7,9,12,11)(21,23,26,22)(24,29,31,27)(25,28,33,30)(32,34,36,35);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,15,16,19)(14,17,18,20)(21,24,26,31)(22,27,23,29)(25,32,33,36)(28,34,30,35);(5,6)(7,9)(8,10)(11,12)(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=gh,d^2=fhi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4302636);