#8930063

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

4754 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=i,f^2=g,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, 8837259);