#147907

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	#1999
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

958 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)(5,6,10,16,12,17,11,36)(7,18,21,51,22,52,37,46)(8,23,26,45,29,49,28,14)(9,30,33,48,35,63,34,15)(13,31,39,57,41,64,40,20)(19,42,54,59,55,65,58,50)(24,38,56,66,53,68,62,67)(25,44,32,47,27,43,61,60);(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)(28,62)(29,53)(32,58)(33,57)(35,64)(36,40)(38,44)(43,68)(45,65)(47,66)(48,51)(49,50)(52,63)(54,61)(60,67);(2,4)(5,8)(6,14)(7,19)(9,32)(10,28)(11,26)(12,29)(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)(39,67)(40,66)(41,68)(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,g^2=i,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, 55103);