#148624

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2033
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

628 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=gi,e^c=eh,f^c=fi,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, 55820);