#494464

order	512
nilpotency class	4
isoclinism	isoclinic class of #33531
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1374 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei,g^c=gi

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, 401660);