#8039024

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7946220);