#8933103

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

4876 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^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, 8840299);