#147694

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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	#2034
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1010 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:

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

Polycyclic presentation:

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

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, 54890);