#421055

order	512
nilpotency class	3
isoclinism	isoclinic class of #417682
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1813 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)(5,6,12,18)(7,19,24,37)(8,25,29,40)(9,30,34,43)(10,17,11,16)(13,23,38,22)(14,27,41,28)(15,32,44,33)(20,47,51,35)(21,36,53,60)(26,39,59,65)(31,46,52,42)(45,49,54,50)(48,66,68,63)(55,57,64,58)(56,67,61,62);(1,2,3,4)(5,7)(6,13)(8,20)(9,31)(10,23)(11,22)(12,24)(14,35)(15,42)(16,37)(17,19)(18,38)(21,32)(25,54)(26,56)(27,50)(28,49)(29,51)(30,60)(33,53)(34,52)(36,43)(39,63)(40,45)(41,47)(44,46)(48,57)(55,62)(58,68)(59,61)(64,67)(65,66);(5,8,12,29)(6,14,18,41)(7,20,24,51)(9,26,34,59)(10,28,11,27)(13,35,38,47)(15,39,44,65)(16,25,17,40)(19,45,37,54)(21,48,53,68)(22,50,23,49)(30,55,43,64)(31,61,52,56)(32,58,33,57)(36,62,60,67)(42,66,46,63);(2,4)(5,9)(6,15)(7,21)(8,26)(10,33)(11,32)(12,34)(13,36)(14,39)(16,30)(17,43)(18,44)(19,46)(20,48)(22,52)(23,31)(24,53)(25,55)(27,58)(28,57)(29,59)(35,62)(37,42)(38,60)(40,64)(41,65)(45,63)(47,67)(49,61)(50,56)(51,68)(54,66)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328251);