#404103

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
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

1298 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)(12,18,31,43)(13,32,34,46)(14,21,15,20)(17,27,40,26)(19,36,44,33)(24,47,51,64)(25,52,54,38)(29,57,59,61)(30,45,35,42)(37,55,63,50)(41,58,66,60)(48,56,65,53)(49,62,69,71)(67,68,72,70);(1,2)(3,6)(4,5)(7,8)(9,11,15,27,16,28,14,26)(10,17,21,39,22,40,20,23)(12,24,35,55,31,51,30,50)(13,25,36,56,34,54,33,53)(18,37,45,64,43,63,42,47)(19,38,46,65,44,52,32,48)(29,49,60,70,59,69,58,68)(41,62,57,72,66,71,61,67);(2,5)(6,8)(9,12)(10,18)(11,24)(13,29)(14,30)(15,35)(16,31)(17,37)(19,41)(20,42)(21,45)(22,43)(23,47)(25,49)(26,50)(27,55)(28,51)(32,61)(33,58)(34,59)(36,60)(38,62)(39,64)(40,63)(44,66)(46,57)(48,67)(52,71)(53,68)(54,69)(56,70)(65,72);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,25)(12,29)(14,33)(15,36)(16,34)(17,38)(18,41)(20,32)(21,46)(22,44)(23,48)(24,49)(26,53)(27,56)(28,54)(30,58)(31,59)(35,60)(37,62)(39,65)(40,52)(42,61)(43,66)(45,57)(47,67)(50,68)(51,69)(55,70)(63,71)(64,72)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^c=hi,f^d=fi,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, 311299);