#473849

order	512
nilpotency class	3
isoclinism	isoclinic class of #454664
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

853 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,23,28,42)(12,29)(13,35)(14,20)(15,21)(16,22)(17,26,43,27)(18,33)(19,38)(24,52,55,40)(25,41,59,69)(30,44,54,66)(31,45)(32,47)(34,46)(36,48)(37,50)(39,49)(51,57,68,58)(53,72,64,67)(56,65,61,71)(60,62)(63,70);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,30)(13,25)(14,27)(15,26)(16,28)(18,44)(19,41)(20,42)(21,23)(22,43)(24,32)(29,52)(31,61)(33,55)(34,54)(35,51)(36,64)(37,58)(38,57)(39,59)(40,46)(45,67)(47,66)(48,65)(49,68)(50,69)(53,62)(56,63)(60,71)(70,72);(3,7)(5,8)(9,12,16,34)(10,18,22,47)(11,24,28,55)(13,36,39,62)(14,33,15,32)(17,40,43,52)(19,48,50,60)(20,46,21,29)(23,44,42,66)(25,56,59,61)(26,54,27,30)(31,38,63,37)(35,45,49,70)(41,67,69,72)(51,71,68,65)(53,58,64,57);(9,13,16,39)(10,19,22,50)(11,25,28,59)(12,31,34,63)(14,38,15,37)(17,41,43,69)(18,45,47,70)(20,49,21,35)(23,51,42,68)(24,53,55,64)(26,58,27,57)(29,60,46,48)(30,61,54,56)(32,62,33,36)(40,65,52,71)(44,67,66,72)

Polycyclic presentation:

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

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, 381045);