#6576479

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,22)(15,19,26,35)(16,20,29,38)(17,21,30,39)(23,32,24,33)(25,34,28,37)(27,36,31,40)(41,43,42,44);(5,7,9,8)(6,10,11,12)(15,23)(16,27)(17,30)(19,32)(20,36)(21,39)(24,26)(29,31)(33,35)(38,40)(41,42)(43,44);(2,4)(13,15)(14,19)(16,28)(17,24)(18,26)(20,37)(21,33)(22,35)(23,30)(25,29)(27,42)(31,41)(32,39)(34,38)(36,44)(40,43);(5,8,9,7)(6,10,11,12)(15,24)(16,27)(17,30)(19,33)(20,36)(21,39)(23,26)(25,28)(29,31)(32,35)(34,37)(38,40);(13,16,18,29)(14,20,22,38)(15,25,26,28)(17,31,30,27)(19,34,35,37)(21,40,39,36)(23,41,24,42)(32,43,33,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483675);