#417735

order	512
nilpotency class	3
isoclinism	isoclinic class of #417732
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2303 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324931);