#148862

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	#2065
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

486 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)(5,6,10,17,11,18,32,45,12,16,33,46,31,44,9,15)(7,13,22,37,23,38,54,66,21,39,55,67,53,65,20,36)(8,24,28,40,29,43,27,47,26,57,30,14,34,41,60,42)(19,48,50,61,51,64,49,68,25,58,52,35,56,62,59,63);(5,7)(6,13)(8,25,26,19)(9,20)(10,22)(11,23)(12,21)(14,35,40,61)(15,36)(16,39)(17,37)(18,38)(24,48,57,58)(27,59,60,49)(28,52,30,50)(29,56,34,51)(31,53)(32,54)(33,55)(41,62,43,64)(42,63,47,68)(44,65)(45,66)(46,67);(2,4)(5,8,11,34,12,26,31,29)(6,14,18,47,16,40,44,42)(7,19,23,56,21,25,53,51)(9,30,10,27,32,28,33,60)(13,35,38,68,39,61,65,63)(15,43,17,24,45,41,46,57)(20,52,22,49,54,50,55,59)(36,64,37,48,66,62,67,58)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=hi,g^2=h,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, 56058);