#147973

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1924
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

674 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)(5,6,10,17,11,18,32,46,12,16,33,44,9,15,30,43)(7,13,22,36,23,37,54,64,21,38,52,65,20,35,51,62)(8,24,28,55,27,40,57,45,26,42,31,58,29,14,39,41)(19,47,49,67,48,60,66,63,25,56,53,68,50,34,59,61);(1,2,3,4)(5,7)(6,13)(8,25,26,19)(9,20)(10,22)(11,23)(12,21)(14,34,40,60)(15,35)(16,38)(17,36)(18,37)(24,47,42,56)(27,50,29,48)(28,53,31,49)(30,51)(32,54)(33,52)(39,66,57,59)(41,61,45,63)(43,62)(44,65)(46,64)(55,67,58,68);(5,8)(6,14)(7,19)(9,29)(10,31)(11,27)(12,26)(13,34)(15,42)(16,40)(17,45)(18,24)(20,50)(21,25)(22,53)(23,48)(28,33)(30,57)(32,39)(35,56)(36,63)(37,47)(38,60)(41,44)(43,55)(46,58)(49,52)(51,66)(54,59)(61,65)(62,67)(64,68)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=i,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=ei,g^c=gi

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, 55169);