#8931054

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

4983 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32);(1,2,5,6)(3,7,8,4)(10,13)(14,15)(17,19,23,20)(18,24,25,21)(22,27,31,28)(26,32,30,29);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(18,25)(19,20)(22,31)(29,32);(1,4)(2,3)(5,7)(6,8)(9,11,12,16)(10,14,13,15)(17,20,23,19)(18,24,25,21)(22,28,31,27)(26,32,30,29);(1,2)(3,4)(5,6)(7,8)(17,21,23,24)(18,20,25,19)(22,29,31,32)(26,28,30,27);(1,4)(2,3)(5,7)(6,8)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)

Polycyclic presentation:

2^9|b^2=h,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8838250);