#7623278

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622979
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#60718	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530474);