#7521756

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2042 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)(13,14,17,19)(15,20,24,30)(16,21,27,31)(18,22,28,32)(23,33,38,41)(25,34,39,42)(26,36,29,35)(37,44,40,43);(1,2)(3,4)(6,10)(11,12)(14,19)(16,26)(18,28)(20,30)(21,35)(23,37)(25,39)(27,29)(31,36)(33,43)(38,40)(41,44);(2,4)(13,15,17,24)(14,20,19,30)(16,23,27,38)(18,25,28,39)(21,33,31,41)(22,34,32,42)(26,37,29,40)(35,43,36,44);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,27)(14,21,19,31)(15,23,24,38)(18,29,28,26)(20,33,30,41)(22,35,32,36)(25,40,39,37)(34,43,42,44);(5,8,9,7)(6,11,10,12)(14,19)(15,24)(21,31)(22,32)(23,38)(25,39)(35,36)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7428952);