#495534

order	512
nilpotency class	4
isoclinism	isoclinic class of #481319
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

806 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)(4,5)(7,8)(9,10,14,20,37,39,35,19,16,22,34,49,15,21,13,33)(11,23,26,17,38,41,25,50,28,52,57,43,27,48,36,42)(12,18,31,46,60,64,58,45,29,44,59,68,32,47,30,61)(24,51,55,40,63,66,54,70,53,71,72,65,56,69,62,67);(9,11,15,38,16,28,37,27)(10,17,39,42,22,43,21,50)(12,24,32,63,29,53,60,56)(13,26,34,25,35,57,14,36)(18,40,64,67,44,65,47,70)(19,48,49,52,33,41,20,23)(30,55,59,54,58,72,31,62)(45,69,68,71,61,66,46,51);(2,5)(6,8)(9,12,16,29)(10,18,22,44)(11,24,28,53)(13,30,35,58)(14,31,34,59)(15,32,37,60)(17,40,43,65)(19,45,33,61)(20,46,49,68)(21,47,39,64)(23,51,52,71)(25,54,36,62)(26,55,57,72)(27,56,38,63)(41,66,48,69)(42,67,50,70);(1,3,4,7)(2,6,5,8)(12,29)(18,44)(24,53)(30,58)(31,59)(32,60)(40,65)(45,61)(46,68)(47,64)(51,71)(54,62)(55,72)(56,63)(66,69)(67,70)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=f,e^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=hi,d^c=di

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, 402730);