#7156224

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021363
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	#61080	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1919 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063420);