#6477738

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

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, 6384934);