#6487508

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62842	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4162 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)(3,6)(4,7)(5,8)(9,10,15,19)(11,20,23,30)(12,17,25,34)(13,18,14,27)(16,28,31,22)(21,36,39,35)(24,32,26,33)(29,37,40,38);(3,4)(6,7)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,37)(25,39)(26,38)(32,36)(33,35)(34,40);(1,3,5,4)(2,6,8,7)(9,12)(10,17)(11,21)(13,24)(14,26)(15,25)(16,29)(18,32)(19,34)(20,35)(22,37)(23,39)(27,33)(28,38)(30,36)(31,40);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,21)(13,26)(14,24)(15,25)(16,29)(18,33)(19,34)(20,36)(22,38)(23,39)(27,32)(28,37)(30,35)(31,40);(2,8)(6,7)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,25,26)(16,30,31,20)(17,32,34,33)(21,37,39,38)(29,36,40,35)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^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,f^b=fi,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394704);