#8107520

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	#36576	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4800 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014716);