#7609928

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7517124);