#355557

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1290 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)(7,17)(8,13)(9,26)(10,30)(11,16)(12,20)(14,28)(15,27)(18,41)(19,44)(21,40)(22,48)(23,51)(24,55)(25,37)(29,56)(31,50)(32,47)(33,45)(34,43)(35,53)(36,52)(38,57)(39,46)(42,58)(49,62)(54,67)(59,66)(60,68)(61,65)(63,64);(5,7)(6,12,16,33)(8,22,25,50)(9,19)(10,20)(11,21)(13,34)(14,32,38,44)(15,17,30,40)(18,24,43,52)(23,49,54,66)(26,39,56,61)(27,45)(28,46)(29,47)(31,55)(35,59)(36,48)(37,41)(42,53,63,68)(51,64)(57,65)(58,67)(60,62);(2,4)(5,8,11,25)(6,13,16,37)(7,18)(9,23,29,54)(10,24,27,52)(12,31)(14,35,38,60)(15,36,30,55)(17,34)(19,42)(20,22)(21,43)(26,51,56,67)(28,53,57,68)(32,58)(33,48)(39,59)(40,41)(44,64)(45,50)(46,49)(47,63)(61,62)(65,66);(5,9,27,57,11,29,10,28)(6,14,30,56,16,38,15,26)(7,19,45,65,21,47,20,46)(8,23,52,68,25,54,24,53)(12,32,40,61,33,44,17,39)(13,35,55,67,37,60,36,51)(18,42,50,66,43,63,22,49)(31,58,41,62,48,64,34,59)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262753);