#7565966

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1986 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7473162);