#497974

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

563 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)(9,10,14,20,15,19,35,47,16,21,36,48,13,31,34,59)(11,22,25,50,24,17,37,39,26,40,54,64,32,41,60,65)(12,18,30,45,29,46,58,70,28,43,57,68,33,44,61,69)(23,49,52,71,53,38,62,63,51,42,66,67,27,55,56,72);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,27)(13,32)(14,25)(15,24)(16,26)(18,42)(19,40)(20,39)(21,41)(22,31)(23,29)(28,53)(30,56)(33,51)(34,60)(35,37)(36,54)(38,44)(43,49)(45,67)(46,55)(47,64)(48,65)(50,59)(52,58)(57,62)(61,66)(63,69)(68,71)(70,72);(1,4,5,3)(2,6,7,8)(12,28)(18,43)(23,51)(27,53)(29,33)(30,57)(38,55)(42,49)(44,46)(45,68)(52,66)(56,62)(58,61)(63,72)(67,71)(69,70);(9,12)(10,18)(11,23)(13,33)(14,30)(15,29)(16,28)(17,38)(19,46)(20,45)(21,43)(22,49)(24,53)(25,52)(26,51)(27,32)(31,44)(34,61)(35,58)(36,57)(37,62)(39,63)(40,42)(41,55)(47,70)(48,68)(50,71)(54,66)(56,60)(59,69)(64,67)(65,72)

Polycyclic presentation:

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

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, 405170);