#148206

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2065
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55402);