#8508896

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4375 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416092);