#8387823

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

5748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gh,e^2=g,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, 8295019);