#494454

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1372 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)(5,6,11,17,34,49,32,47,12,18,10,16,9,15,33,48)(7,19,24,39,21,51,35,13,25,40,22,38,23,50,36,41)(8,26,30,61,29,46,62,44,31,43,28,60,27,14,42,45)(20,53,56,67,54,66,63,59,58,68,55,37,57,52,64,65);(1,2,3,4)(5,7,12,25)(6,13,18,41)(8,20,31,58)(9,21,34,23)(10,35,11,36)(14,37,46,67)(15,38,49,39)(16,50,17,51)(19,48,40,47)(22,33,24,32)(26,59,43,65)(27,54,29,57)(28,63,30,64)(42,56,62,55)(44,53,45,68)(52,61,66,60);(5,8)(6,14)(7,20)(9,27)(10,30)(11,28)(12,31)(13,37)(15,43)(16,45)(17,44)(18,46)(19,52)(21,54)(22,56)(23,57)(24,55)(25,58)(26,49)(29,34)(32,42)(33,62)(35,64)(36,63)(38,65)(39,59)(40,66)(41,67)(47,60)(48,61)(50,68)(51,53);(5,9,12,34)(6,15,18,49)(7,21,25,23)(8,27,31,29)(10,32,11,33)(13,38,41,39)(14,43,46,26)(16,47,17,48)(19,51,40,50)(20,54,58,57)(22,36,24,35)(28,62,30,42)(37,65,67,59)(44,61,45,60)(52,53,66,68)(55,64,56,63)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401650);