#8614118

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

5078 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:

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

Polycyclic presentation:

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

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, 8521314);