#8093293

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	#35533	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4902 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)(11,15)(12,13)(14,16)(17,18)(19,25)(20,30)(21,22)(23,24)(26,28)(27,31)(29,32);(2,5)(3,7)(10,12)(14,15)(18,24)(21,27)(25,32)(26,30);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,19,23,29)(18,25,24,32)(20,22,28,31)(21,26,27,30);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,23,28)(18,26,24,30)(19,22,29,31)(21,25,27,32);(17,21,23,27)(18,22,24,31)(19,26,29,30)(20,25,28,32);(17,22)(18,27)(19,28)(20,29)(21,24)(23,31)(25,30)(26,32)

Polycyclic presentation:

2^9|c^2=gh,d^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, 8000489);