#6482267

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470922
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1674 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:

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

Polycyclic presentation:

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

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, 6389463);