#7603082

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7510278);