#147900

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1970
prime radical	C₂²×#508	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)(7,19,23,49)(8,25,29,14)(9,17)(10,33)(11,35)(12,16)(13,21,39,57)(15,31)(18,34)(20,42)(22,51,58,41)(24,46,36,40)(26,38)(27,47,61,44)(28,43,62,50)(30,45,37,52)(32,48)(53,56)(54,68)(55,64)(59,67)(60,65)(63,66);(2,4)(5,7)(6,13)(8,26)(9,22)(10,21)(11,36)(12,23)(14,42)(15,46)(16,39)(17,49)(18,51)(19,48)(20,28)(24,34)(25,53)(27,59)(29,60)(30,63)(31,57)(32,58)(33,40)(35,41)(37,54)(38,44)(43,66)(45,64)(47,65)(50,68)(52,67)(55,61)(56,62);(5,8)(6,14)(7,20)(9,28)(10,27)(11,37)(12,29)(13,38)(15,47)(16,25)(17,50)(18,52)(19,53)(21,26)(22,54)(23,56)(24,59)(30,34)(31,61)(32,62)(33,44)(35,45)(36,55)(39,65)(40,67)(41,68)(42,49)(43,48)(46,64)(51,66)(57,60)(58,63)

Polycyclic presentation:

2^9|a^2=g,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dhi,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, 55096);