#148944

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

450 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)(7,12,18,33)(8,13,22,37)(9,15)(10,16)(11,14)(17,32)(19,34,49,61)(20,35,29,45)(21,36)(23,38,54,58)(24,39,30,46)(25,42,52,67)(26,43,56,68)(27,44)(28,40)(31,41)(47,60)(48,59)(50,63)(51,64)(53,65)(55,66)(57,62);(5,7)(6,12,14,33)(8,21,22,53)(9,25)(10,29)(11,18)(13,36)(15,42,40,67)(16,45,41,35)(17,23,47,54)(19,27)(20,31)(24,55,30,50)(26,48,56,51)(28,52)(32,58)(34,62,61,44)(37,65)(38,60)(39,66)(43,59)(46,63)(49,57)(64,68);(5,8)(6,13,14,37)(7,17,18,47)(9,26)(10,30)(11,22)(12,32)(15,43,40,68)(16,46,41,39)(19,50,49,55)(20,51,29,48)(21,52,53,25)(23,27)(24,31)(28,56)(33,60)(34,63)(35,64)(36,42)(38,62,58,44)(45,59)(54,57)(61,66)(65,67)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=di,f^b=fh,g^b=gi,h^b=hi,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, 56140);