#8180448

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

5015 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:

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

Polycyclic presentation:

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

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, 8087644);