#5083698

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1860 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,17)(18,20)(21,22,26,30)(23,31,33,27)(24,34,35,28)(25,32,36,29);(6,9)(7,12)(13,14)(15,19)(16,17)(18,20)(21,23,26,33)(22,27,30,31)(24,29,35,32)(25,34,36,28);(6,9)(10,11)(14,17)(18,19)(21,23)(22,27)(24,32)(25,34)(26,33)(28,36)(29,35)(30,31);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,28)(23,32)(25,31)(26,35)(27,36)(29,33)(30,34);(5,6)(7,10)(8,9)(11,12)(13,15,16,20)(14,18,17,19)(21,25,26,36)(22,29,30,32)(23,28,33,34)(24,31,35,27)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4990894);