#6524946

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1850 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)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22,27,35,26,29,25,30)(23,28,36,33,31,24,32,34);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,28)(24,29)(25,32)(26,31)(27,36)(30,34)(33,35);(6,11)(7,13)(8,15)(16,20)(21,24,26,28)(22,23,29,31)(25,34,27,33)(30,36,35,32);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(22,29)(23,31)(30,35)(32,36);(7,13)(10,17)(14,19)(16,20)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6432142);