#4309093

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

1869 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)(5,6,8,10)(7,9,11,12)(13,14,17,20)(15,19,16,18)(22,27)(24,29)(25,33)(30,36);(13,14,17,20)(15,18,16,19)(21,22,26,27)(23,29,31,24)(25,28,33,34)(30,35,36,32);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,23,26,31)(22,24,27,29)(25,30,33,36)(28,32,34,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,16)(14,18,20,19)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,35)(34,36);(5,7,8,11)(6,9,10,12)(13,16,17,15)(14,19,20,18)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=fh,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, 4216289);