#8934393

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

5299 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841589);