#8016644

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

4973 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,e^2=gi,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7923840);