#498036

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405232);