#147906

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

966 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)(7,18)(8,23,28,49)(9,30)(10,16)(11,36)(12,17)(13,20)(14,26,45,29)(15,34)(19,42,55,65)(21,51)(22,52)(24,38,53,67)(25,44,27,43)(31,39)(32,47,61,60)(33,48)(35,63)(37,46)(40,64)(41,57)(50,54,59,58)(56,66,62,68);(2,4)(5,7)(6,13)(8,24)(9,31)(10,21)(11,37)(12,22)(14,42)(15,46)(16,39)(17,41)(18,30)(19,27)(20,34)(23,59)(25,55)(26,56)(28,53)(29,62)(32,58)(33,57)(35,64)(36,40)(38,44)(43,67)(45,65)(47,66)(48,51)(49,50)(52,63)(54,61)(60,68);(5,8)(6,14)(7,19)(9,32)(10,29)(11,26)(12,28)(13,38)(15,47)(16,49)(17,45)(18,50)(20,56)(21,58)(22,55)(23,36)(24,31)(25,33)(27,34)(30,44)(35,61)(37,54)(39,68)(40,66)(41,67)(42,46)(43,63)(48,60)(51,65)(52,59)(53,64)(57,62)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=dh,f^b=fh,h^b=hi,d^c=dhi,e^c=ei,f^c=fh,g^c=gi,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, 55102);