#7021383

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021375
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=f,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

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, 6928579);