#461276

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 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)(7,19,23,37)(8,24)(9,15)(10,16)(11,17)(12,18)(13,22,38,34)(14,27)(20,48,50,35)(21,47,53,63)(25,39,54,61)(26,55)(28,42)(29,41)(30,40)(31,44)(32,45)(33,46)(36,52,64,59)(43,57)(49,67,58,60)(51,65,68,62)(56,66);(5,7,12,23)(6,13)(8,25,29,54)(9,21,33,53)(10,22,11,34)(14,39)(15,36)(16,19)(17,37)(18,38)(20,27,50,28)(24,48)(26,51,57,68)(30,58,56,49)(31,52,32,59)(35,41)(40,65)(42,61)(43,60)(44,47)(45,63)(46,64)(55,67)(62,66);(5,8,12,29)(6,14,18,42)(7,20,23,50)(9,30,33,56)(10,28,11,27)(13,35,38,48)(15,43,46,55)(16,24,17,41)(19,39,37,61)(21,51,53,68)(22,54,34,25)(26,32,57,31)(36,62,64,65)(40,45,66,44)(47,60,63,67)(49,59,58,52);(2,4)(5,9,11,32,12,33,10,31)(6,15,17,45,18,46,16,44)(7,21,34,59,23,53,22,52)(8,26,28,56,29,57,27,30)(13,36,37,63,38,64,19,47)(14,40,24,55,42,66,41,43)(20,49,54,68,50,58,25,51)(35,60,39,65,48,67,61,62)

Polycyclic presentation:

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

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, 368472);