#476994

order	512
nilpotency class	3
isoclinism	isoclinic class of #476112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1036 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384190);