#7520782

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fi,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,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, 7427978);