#8094771

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

4921 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8001967);