#8388797

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

5030 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8295993);