#8934363

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

4856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=gh,f^2=h,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, 8841559);