#8924948

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

4953 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)(10,13)(11,15)(17,18,22,20)(19,23,26,27)(21,24,29,28)(25,30,31,32);(1,3,4,7)(2,5,6,8)(9,10,14,13)(11,16,15,12)(18,20)(23,27)(24,28)(30,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(21,29)(24,28)(25,31)(30,32);(9,10,14,13)(11,12,15,16)(17,19,22,26)(18,23,20,27)(21,25,29,31)(24,30,28,32);(1,3)(2,5)(4,7)(6,8)(9,12,14,16)(10,15,13,11)(17,20,22,18)(19,23,26,27)(21,28,29,24)(25,30,31,32);(1,2)(3,5)(4,6)(7,8)(9,13,14,10)(11,16,15,12)(17,21,22,29)(18,24,20,28)(19,25,26,31)(23,30,27,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832144);