#8436589

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	#43023	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4922 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=gh,f^2=g,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, 8343785);