#472935

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egh,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380131);