#7523124

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,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, 7430320);