#495571

order	512
nilpotency class	4
isoclinism	isoclinic class of #481349
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

796 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)(9,10,14,20,16,21,15,37)(11,22,25,49,26,47,36,50)(12,18,29,46,27,44,30,61)(13,31,33,39,35,62,34,19)(17,38,41,24,43,54,42,32)(23,48,53,69,51,68,60,70)(28,56,58,64,55,72,59,45)(40,63,66,52,65,71,67,57);(9,11,13,32,15,25,34,38,16,26,35,24,14,36,33,54)(10,17,39,22,20,42,62,50,21,43,19,47,37,41,31,49)(12,23,28,57,30,53,59,63,27,51,55,52,29,60,58,71)(18,40,64,48,46,67,72,70,44,65,45,68,61,66,56,69);(2,5)(6,8)(9,12,16,27)(10,18,21,44)(11,23,26,51)(13,28,35,55)(14,29,15,30)(17,40,43,65)(19,45,39,64)(20,46,37,61)(22,48,47,68)(24,52,32,57)(25,53,36,60)(31,56,62,72)(33,58,34,59)(38,63,54,71)(41,66,42,67)(49,69,50,70);(1,3,4,7)(2,6,5,8)(12,27)(18,44)(23,51)(28,55)(29,30)(40,65)(45,64)(46,61)(48,68)(52,57)(53,60)(56,72)(58,59)(63,71)(66,67)(69,70)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402767);