#147888

order	512
nilpotency class	4
isoclinism	isoclinic class of #140598
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,g^b=gi,h^b=hi,d^c=dhi,e^c=eh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281717, #72281903, #72282014, …
maximal normal subgroup of	#72281717, #72281903, #72282014, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55084);