#4422596

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1960 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=f,d^2=fg,e^2=fh,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, 4329792);