#7523120

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3010	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=f,e^2=fi,g^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,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7430316);