#8131498

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

4954 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8038694);