#8700464

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	#59112	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4630 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)(3,6)(4,7)(5,8)(9,10,14,12)(11,15,13,16);(1,3)(2,4)(5,7)(6,8)(9,11,14,13)(10,15,12,16)(17,18)(19,26)(20,28)(21,23)(22,25)(24,29)(27,31)(30,32);(9,11,14,13)(10,16,12,15)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32);(1,2,5,8)(3,6,7,4)(9,12,14,10)(11,15,13,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,15,12,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,31)(28,29)(30,32);(1,4,5,6)(2,7,8,3)(17,21,22,19)(18,23,25,26)(20,27,29,30)(24,32,28,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=gh,e^2=h,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, 8607660);