#4085080

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

1782 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=fhi,d^2=f,e^2=g,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, 3992276);