#493550

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3134 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)(5,6,9,15,12,17,11,33)(7,18,20,45,22,46,34,42)(8,23,26,41,27,39,24,14)(10,29,28,43,32,60,31,16)(13,30,36,52,38,61,37,21)(19,47,50,66,51,65,48,44)(25,40,57,54,58,64,55,59)(35,49,53,67,63,68,62,56);(5,7)(6,13)(8,19)(9,20)(10,30)(11,34)(12,22)(14,35)(15,36)(16,42)(17,38)(18,29)(21,31)(23,53)(24,48)(25,56)(26,50)(27,51)(28,52)(32,61)(33,37)(39,62)(40,47)(41,63)(43,45)(44,59)(46,60)(49,57)(54,66)(55,68)(58,67)(64,65);(2,4)(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,27)(13,35)(15,39)(16,40)(17,41)(18,44)(20,48)(21,49)(22,51)(23,33)(28,55)(29,59)(30,56)(31,57)(32,58)(34,50)(36,62)(37,53)(38,63)(42,47)(43,64)(45,65)(46,66)(52,68)(54,60)(61,67);(5,9,12,11)(6,15,17,33)(7,20,22,34)(8,24,27,26)(10,28,32,31)(13,36,38,37)(14,39,41,23)(16,29,43,60)(18,45,46,42)(19,48,51,50)(21,30,52,61)(25,55,58,57)(35,62,63,53)(40,59,64,54)(44,65,66,47)(49,56,68,67)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,h^b=hi,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, 400746);