#464649

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1457 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)(9,14)(10,28)(11,16)(12,20)(15,31)(18,44)(19,50)(21,46)(22,57)(23,42)(24,38)(25,39)(26,43)(27,41)(29,37)(30,40)(32,54)(33,52)(34,56)(35,48)(36,58)(45,53)(47,62)(49,61)(51,60)(55,63)(59,65)(64,68)(66,67);(5,7)(6,12,16,34)(8,18)(9,19)(10,20)(11,21)(13,35,38,32)(14,33,41,63)(15,17,28,46)(22,47)(23,48)(24,49)(25,51)(26,52)(27,53)(29,54)(30,55)(31,56)(36,64,57,60)(37,44,42,61)(39,66,65,62)(40,50,43,45)(58,67)(59,68);(2,4)(5,8,11,24)(6,13,16,38)(7,18,21,49)(9,25,27,59)(10,29,31,23)(12,32,34,35)(14,39,41,65)(15,42,28,37)(17,44,46,61)(19,51,53,68)(20,54,56,48)(22,30,58,26)(33,62,63,66)(36,43,57,40)(45,64,50,60)(47,55,67,52);(5,9,11,27)(6,14,16,41)(7,19,21,53)(8,22,24,58)(10,30,31,26)(12,33,34,63)(13,36,38,57)(15,43,28,40)(17,45,46,50)(18,47,49,67)(20,55,56,52)(23,59,29,25)(32,60,35,64)(37,65,42,39)(44,62,61,66)(48,68,54,51)

Polycyclic presentation:

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

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, 371845);