#7519493

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

1734 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)(13,14,19,25)(15,27,31,21)(16,32,29,40)(17,36,28,44)(18,26,20,24)(22,34,41,35)(23,33,42,37)(30,39,38,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,33)(17,34)(18,30)(19,31)(20,38)(22,36)(23,32)(24,39)(25,27)(26,43)(28,35)(29,37)(40,42)(41,44);(13,16,19,29)(14,22,25,41)(15,28,31,17)(18,35,20,34)(21,23,27,42)(24,32,26,40)(30,37,38,33)(36,39,44,43);(2,4)(13,17,19,28)(14,23,25,42)(15,29,31,16)(18,37,20,33)(21,22,27,41)(24,36,26,44)(30,35,38,34)(32,39,40,43);(5,8,9,7)(6,11,12,10)(13,18,19,20)(14,24,25,26)(15,30,31,38)(16,34,29,35)(17,37,28,33)(21,39,27,43)(22,40,41,32)(23,36,42,44)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=i,e^2=f,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=cg,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426689);