#8701738

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	#59952	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4837 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,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=dghi,f^d=fgi,f^e=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, 8608934);