#6510839

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37450	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2372 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=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, 6418035);