#398064

order	512
nilpotency class	3
isoclinism	isoclinic class of #394358
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1275 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305260);