#404120

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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:

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

Polycyclic presentation:

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

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, 311316);