#149107

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1923
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

430 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:

(1,2,4,3)(5,6,11,18,10,15,32,43,12,17,33,45,9,16,31,44)(7,19,23,47,21,13,34,38,24,36,52,61,22,37,51,62)(8,25,30,53,28,14,39,41,29,40,57,64,27,42,56,65)(20,35,50,60,48,59,58,68,26,54,55,67,49,46,63,66);(1,3,4,2)(5,7)(6,13)(8,26)(9,22)(10,21)(11,23)(12,24)(14,35)(15,36)(16,19)(17,37)(18,38)(20,29)(25,46)(27,48)(28,49)(30,55)(31,51)(32,34)(33,52)(39,63)(40,59)(41,60)(42,54)(43,61)(44,47)(45,62)(50,57)(53,66)(56,58)(64,68)(65,67);(2,3)(5,8,9,27,12,29,10,28)(6,14,15,40,17,42,16,25)(7,20,21,48,24,26,22,49)(11,30,31,56,33,57,32,39)(13,35,19,46,37,54,36,59)(18,41,43,64,45,65,44,53)(23,50,34,58,52,55,51,63)(38,60,47,66,62,67,61,68)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56303);