#8688273

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

4845 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8595469);