#148207

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	#2054
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

684 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)(5,6,10,17,31,45,32,46,12,16,9,15,11,18,34,48)(7,13,22,38,55,64,56,65,21,40,20,37,23,39,58,67)(8,24,28,43,35,47,30,14,26,42,27,49,29,44,33,41)(19,50,52,62,59,66,54,36,25,60,51,68,53,63,57,61);(5,7)(6,13)(8,25,26,19)(9,20)(10,22)(11,23)(12,21)(14,36,41,61)(15,37)(16,40)(17,38)(18,39)(24,50,42,60)(27,52,28,51)(29,59,35,53)(30,57,33,54)(31,55)(32,56)(34,58)(43,62,49,68)(44,63,47,66)(45,64)(46,65)(48,67);(2,4)(5,8)(6,14)(7,19)(9,30)(10,33)(11,35)(12,26)(13,36)(15,44)(16,41)(17,47)(18,49)(20,54)(21,25)(22,57)(23,59)(24,46)(27,32)(28,34)(29,31)(37,63)(38,66)(39,68)(40,61)(42,48)(43,45)(50,65)(51,56)(52,58)(53,55)(60,67)(62,64)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,g^2=hi,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, 55403);