#454684

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

1743 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)(9,31,35,45)(10,17)(11,16)(12,18)(13,22)(14,29)(15,34,46,33)(20,48)(21,37,55,65)(23,39)(24,38)(26,56)(27,58,60,41)(28,43)(30,42)(32,44,59,66)(36,51)(40,50)(47,54,64,53)(49,68,61,63)(52,67,57,62);(1,2)(3,4)(5,7,12,24)(6,13,18,39)(8,26,30,50)(9,21,35,55)(10,23,11,22)(14,40,43,56)(15,37,46,65)(16,38,17,19)(20,29,51,28)(25,36,42,48)(27,57,60,52)(31,47,45,64)(32,61,59,49)(33,54,34,53)(41,63,58,68)(44,62,66,67);(2,4)(5,8,12,30)(6,14,18,43)(7,20,24,51)(9,32,35,59)(10,29,11,28)(13,36,39,48)(15,44,46,66)(16,25,17,42)(19,40,38,56)(21,52,55,57)(22,50,23,26)(27,34,60,33)(31,41,45,58)(37,63,65,68)(47,67,64,62)(49,54,61,53);(5,9)(6,15)(7,21)(8,27)(10,34)(11,33)(12,35)(13,37)(14,41)(16,45)(17,31)(18,46)(19,47)(20,49)(22,54)(23,53)(24,55)(25,44)(26,57)(28,59)(29,32)(30,60)(36,62)(38,64)(39,65)(40,63)(42,66)(43,58)(48,67)(50,52)(51,61)(56,68)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361880);