#4310618

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1973 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)(5,6,10,9)(7,12,11,8)(13,14)(15,16)(17,20)(18,19)(21,22)(23,28)(24,26)(25,34)(27,30)(29,36)(31,35)(32,33);(13,15)(14,18)(16,20)(17,19)(21,23,27,25)(22,28,30,34)(24,33,29,31)(26,32,36,35);(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,24,27,29)(22,26,30,36)(23,31,25,33)(28,35,34,32);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,16)(14,19)(15,20)(17,18)(21,25,27,23)(22,28,30,34)(24,33,29,31)(26,35,36,32);(5,9,10,6)(7,12,11,8)(21,26)(22,29)(23,32)(24,30)(25,35)(27,36)(28,31)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4217814);