#6463034

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60644	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10,13,17,14,19,26,18)(11,20,22,15,23,28,36,29)(12,16,25,31,24,30,37,32)(21,33,35,27,34,39,40,38);(3,4)(6,8)(9,10)(11,15)(12,16)(13,18)(14,19)(17,26)(20,22)(21,27)(23,29)(24,30)(25,32)(28,36)(31,37)(33,35)(34,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11,13,22,14,23,26,36)(10,15,18,20,19,29,17,28)(12,21,25,35,24,34,37,40)(16,27,32,33,30,38,31,39);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,21)(13,25)(14,24)(15,27)(17,31)(18,32)(19,30)(20,33)(22,35)(23,34)(26,37)(28,39)(29,38)(36,40);(12,24)(16,30)(21,34)(25,37)(27,38)(31,32)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,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, 6370230);