#147994

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 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:

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

Polycyclic presentation:

2^9|a^2=g,b^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,f^b=fi,d^c=di,e^c=ei

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, 55190);