#4316485

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

2294 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)(10,12)(15,19)(18,20)(21,22,25,29)(23,27,31,26)(24,32,34,28)(30,35,33,36);(14,17)(18,20)(21,23,25,31)(22,26,29,27)(24,33,34,30)(28,36,32,35);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,36)(32,35)(33,34);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,22)(23,26)(24,28)(25,29)(27,31)(30,35)(32,34)(33,36);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,24,25,34)(22,28,29,32)(23,30,31,33)(26,35,27,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4223681);