#365772

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1234 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=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, 272968);