#4084803

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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

1776 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

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, 3991999);