#4343737

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=g,d^2=fgi,e^2=fgh,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, 4250933);