#4320402

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

2050 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)(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,30)(24,25)(26,27)(28,33)(29,35)(31,32)(34,36);(7,11)(9,12)(13,15,16,19)(14,17,18,20)(22,27)(25,29)(28,30)(32,36);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,17)(16,19)(18,20)(21,23,26,33)(22,28,27,30)(24,34,35,31)(25,32,29,36);(1,3,4,2)(13,14)(15,17)(16,18)(19,20)(21,24)(22,29)(23,31)(25,27)(26,35)(28,32)(30,36)(33,34);(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,25,26,29)(22,24,27,35)(23,32,33,36)(28,34,30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=fg,e^2=f,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, 4227598);