#497464

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3417	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

947 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)(5,6,12,18,13,9,11,17)(7,19,24,21,25,48,23,46)(8,26,30,15,31,45,29,27)(10,34,38,33,39,36,37,16)(14,35,43,54,32,41,42,22)(20,49,52,47,53,67,51,68)(28,58,62,56,63,44,61,60)(40,64,66,59,57,50,55,65);(5,7)(6,14)(8,20)(9,32)(10,35)(11,23)(12,24)(13,25)(15,40)(16,46)(17,42)(18,43)(19,34)(21,33)(22,37)(26,55)(27,57)(28,59)(29,51)(30,52)(31,53)(36,48)(38,54)(39,41)(44,67)(45,66)(47,56)(49,58)(50,62)(60,68)(61,64)(63,65);(2,4)(5,8,13,31)(6,15,9,27)(7,20,25,53)(10,28,39,63)(11,30,12,29)(14,40,32,57)(16,44,33,58)(17,45,18,26)(19,47,48,68)(21,49,46,67)(22,50,54,64)(23,52,24,51)(34,56,36,60)(35,59,41,65)(37,62,38,61)(42,66,43,55);(5,9)(6,13)(7,21)(8,27)(10,36)(11,18)(12,17)(14,41)(15,31)(16,38)(19,23)(20,49)(22,43)(24,48)(25,46)(26,30)(28,60)(29,45)(32,35)(33,37)(34,39)(40,65)(42,54)(44,61)(47,52)(50,55)(51,68)(53,67)(56,63)(57,59)(58,62)(64,66)

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,f^c=fi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404660);