#7149007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 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:

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

Polycyclic presentation:

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

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, 7056203);