#427882

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1196 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335078);