#8687351

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

4844 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8594547);