#455482

order	512
nilpotency class	3
isoclinism	isoclinic class of #454110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1101 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:

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

Polycyclic presentation:

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

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, 362678);