#368893

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1652 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:

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

Polycyclic presentation:

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

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, 276089);