#8093264

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	#35563	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5032 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:

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

Polycyclic presentation:

2^9|c^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000460);