#6457715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2)(3,7)(4,8)(5,6)(9,10,15,21,16,22,14,20)(11,23,26,17,28,41,27,42)(12,29,32,46,34,45,33,18)(13,19,36,48,38,50,37,49)(24,53,30,51,56,39,55,43)(25,52,35,40,58,47,57,66)(31,59,63,70,64,69,62,44)(54,68,60,71,61,65,72,67);(2,6)(7,8)(9,11)(10,17)(12,30)(13,35)(14,26)(15,27)(16,28)(18,43)(19,47)(20,41)(21,23)(22,42)(24,32)(25,36)(29,39)(31,61)(33,56)(34,55)(37,58)(38,57)(40,48)(44,68)(45,53)(46,51)(49,66)(50,52)(54,64)(59,67)(60,63)(62,72)(65,70)(69,71);(1,3,5,4)(2,7,6,8)(9,12)(10,18,22,46)(11,24,28,56)(13,31)(14,33)(15,32)(16,34)(17,39)(19,44,50,70)(20,45,21,29)(23,51)(25,54,58,61)(26,30,27,55)(35,60,57,72)(36,63)(37,62)(38,64)(40,65)(41,43)(42,53)(47,67)(48,59,49,69)(52,71)(66,68);(1,4,5,3)(2,8,6,7)(9,13,14,37,16,38,15,36)(10,19,20,49,22,50,21,48)(11,25,27,57,28,58,26,35)(12,31,33,62,34,64,32,63)(17,40,23,52,42,66,41,47)(18,44,45,69,46,70,29,59)(24,54,55,72,56,61,30,60)(39,65,51,71,53,68,43,67);(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,32,34,33)(13,36,38,37)(17,41,42,23)(18,29,46,45)(19,48,50,49)(24,55,56,30)(25,57,58,35)(31,62,64,63)(39,51,53,43)(40,52,66,47)(44,69,70,59)(54,60,61,72)(65,67,68,71)

Polycyclic presentation:

2^9|a^2=h,c^2=fgi,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,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, 6364911);