#394037

order	512
nilpotency class	3
isoclinism	isoclinic class of #394034
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3586	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301233);