#494048

order	512
nilpotency class	4
isoclinism	isoclinic class of #481284
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

940 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)(9,10,14,19,15,21,35,20)(11,16,25,39,23,37,52,40)(12,26,28,44,29,58,34,57)(13,30,32,18,33,45,61,46)(17,31,41,27,43,59,42,60)(22,47,50,66,48,68,56,69)(24,53,55,38,51,67,72,65)(36,54,63,49,62,71,64,70);(2,5)(6,8)(9,11)(10,16)(12,22)(13,24)(14,25)(15,23)(17,36)(18,38)(19,39)(20,40)(21,37)(26,47)(27,49)(28,50)(29,48)(30,53)(31,54)(32,55)(33,51)(34,56)(35,52)(41,63)(42,64)(43,62)(44,66)(45,67)(46,65)(57,69)(58,68)(59,71)(60,70)(61,72);(9,12,15,29)(10,17,21,43)(11,22,23,48)(13,31,33,59)(14,34,35,28)(16,36,37,62)(18,44,46,57)(19,42,20,41)(24,54,51,71)(25,56,52,50)(26,30,58,45)(27,32,60,61)(38,66,65,69)(39,64,40,63)(47,53,68,67)(49,55,70,72);(1,3,4,7)(2,6,5,8)(11,23)(16,37)(22,48)(24,51)(25,52)(36,62)(38,65)(39,40)(47,68)(49,70)(50,56)(53,67)(54,71)(55,72)(63,64)(66,69)

Polycyclic presentation:

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

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, 401244);