#494299

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1046 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)(7,20)(8,26)(9,16)(10,18)(11,17)(12,34)(13,19)(14,23)(15,29)(21,51)(22,39)(24,40)(25,50)(27,43)(28,44)(30,62)(31,46)(32,38)(33,48)(35,47)(36,61)(37,52)(41,58)(42,63)(45,64)(49,54)(53,60)(55,67)(56,65)(57,66)(59,68);(1,3,4,2)(5,7)(6,14)(8,21)(9,22)(10,32)(11,23)(12,35)(13,25)(15,37)(16,38)(17,47)(18,20)(19,41)(24,33)(26,60)(27,52)(28,57)(29,53)(30,56)(31,58)(34,40)(36,55)(39,46)(42,65)(43,49)(44,51)(45,67)(48,50)(54,63)(59,64)(61,66)(62,68);(5,8)(6,15)(7,21)(9,27)(10,29)(11,28)(12,36)(13,30)(14,37)(16,42)(17,26)(18,43)(19,45)(20,49)(22,52)(23,57)(24,59)(25,56)(31,63)(32,53)(33,64)(34,44)(35,55)(38,65)(39,68)(40,51)(41,67)(46,62)(47,60)(48,61)(50,66)(54,58);(5,9,13,12)(6,16,19,34)(7,22,25,35)(8,27,30,36)(10,31,33,11)(14,38,41,40)(15,42,45,44)(17,18,46,48)(20,39,50,47)(21,52,56,55)(23,32,58,24)(26,43,62,61)(28,29,63,64)(37,65,67,51)(49,68,66,60)(53,54,59,57)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fhi,h^c=hi

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, 401495);