#494295

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1034 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)(5,6,10,16,12,17,11,33)(7,18,22,48,20,47,34,43)(8,23,25,55,26,56,35,44)(9,27,30,45,32,58,31,15)(13,28,38,53,37,59,39,21)(14,29,40,57,42,60,41,24)(19,36,51,63,49,61,52,64)(46,65,67,68,66,54,62,50);(2,4)(5,7,10,22,12,20,11,34)(6,13,16,38,17,37,33,39)(8,19,25,51,26,49,35,52)(9,28,30,53,32,59,31,21)(14,36,40,63,42,61,41,64)(15,43,27,18,45,48,58,47)(23,46,55,67,56,66,44,62)(24,54,29,50,57,65,60,68);(5,8)(6,14)(7,19)(9,29)(10,25)(11,35)(12,26)(13,36)(15,44)(16,40)(17,42)(18,46)(20,49)(21,54)(22,51)(23,27)(24,31)(28,50)(30,57)(32,60)(33,41)(34,52)(37,61)(38,63)(39,64)(43,62)(45,55)(47,66)(48,67)(53,65)(56,58)(59,68);(7,20)(13,37)(18,47)(19,49)(21,53)(22,34)(28,59)(36,61)(38,39)(43,48)(46,66)(50,68)(51,52)(54,65)(62,67)(63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323138
maximal normal subgroup of	#72323138
maximal quotient of	#72323064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401491);