#391210

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1045 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,g^b=gi,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, 298406);