#7148505

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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

1691 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)(9,10,14,20,15,19,13,28)(11,21,23,16,24,32,29,17)(12,18,26,37,27,35,30,36)(22,38,39,31,40,34,25,33);(2,5)(3,7)(9,11)(10,16)(12,25)(13,29)(14,23)(15,24)(17,19)(18,34)(20,32)(21,28)(22,26)(27,39)(30,40)(31,36)(33,37)(35,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,22)(13,23)(14,29)(15,24)(16,19)(18,33)(20,32)(21,28)(25,26)(27,40)(30,39)(31,35)(34,37)(36,38);(2,5)(6,8)(9,12,15,27)(10,18,19,35)(11,22,24,40)(13,30,14,26)(16,31,17,33)(20,37,28,36)(21,38,32,34)(23,39,29,25);(1,3,4,7)(2,6,5,8)(10,19)(16,17)(18,35)(20,28)(21,32)(31,33)(34,38)(36,37)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7055701);