#8458061

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

5064 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8365257);