#8230801

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	#59408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4967 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:

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

Polycyclic presentation:

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

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, 8137997);