#6578120

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#32913	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2057 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)(5,6,9,10)(7,11,8,12)(13,14,18,22)(15,19,26,35)(16,27,30,20)(17,21,31,38)(23,32,25,34)(24,40,41,33)(28,37,29,36)(39,44,42,43);(6,10)(11,12)(13,15,18,26)(14,19,22,35)(16,24,30,41)(17,23,31,25)(20,33,27,40)(21,32,38,34)(28,42,29,39)(36,44,37,43);(2,4)(15,23)(16,28)(17,31)(19,32)(20,36)(21,38)(24,41)(25,26)(27,37)(29,30)(33,40)(34,35);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,24)(17,29)(18,30)(19,33)(21,37)(22,27)(23,39)(25,42)(26,41)(28,31)(32,43)(34,44)(35,40)(36,38);(5,8,9,7)(6,12,10,11)(15,23)(16,29)(17,31)(19,32)(20,37)(21,38)(25,26)(27,36)(28,30)(34,35)(39,42)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485316);