#8184861

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

4938 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=gh,e^2=h,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, 8092057);