#464067

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1407 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)(5,6,12,18)(7,19,24,36)(8,14,29,42)(9,15,32,45)(10,17,11,16)(13,23,37,22)(20,38,50,61)(21,51,54,47)(25,34,49,62)(26,56,57,39)(27,41,28,40)(30,43,58,65)(31,46,33,44)(35,55,64,53)(48,63,59,68)(52,60,67,66);(5,7,11,23,12,24,10,22)(6,13,17,36,18,37,16,19)(8,25,28,50,29,49,27,20)(9,21,31,53,32,54,33,55)(14,38,41,62,42,61,40,34)(15,35,44,51,45,64,46,47)(26,52,30,59,57,67,58,48)(39,63,43,66,56,68,65,60);(2,4)(5,8,12,29)(6,14,18,42)(7,20,24,50)(9,30,32,58)(10,28,11,27)(13,34,37,62)(15,43,45,65)(16,41,17,40)(19,38,36,61)(21,52,54,67)(22,25,23,49)(26,33,57,31)(35,63,64,68)(39,46,56,44)(47,66,51,60)(48,55,59,53);(5,9)(6,15)(7,21)(8,26)(10,33)(11,31)(12,32)(13,35)(14,39)(16,46)(17,44)(18,45)(19,47)(20,48)(22,55)(23,53)(24,54)(25,52)(27,58)(28,30)(29,57)(34,60)(36,51)(37,64)(38,63)(40,65)(41,43)(42,56)(49,67)(50,59)(61,68)(62,66)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,e^2=i,g^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,g^c=gi,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, 371263);