#8341811

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
prime radical	#59808	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4818 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8249007);