#8094917

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

4894 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)(10,13)(14,16)(17,18,23,21)(19,25,30,24)(20,27,22,26)(28,31,29,32);(1,2,5,3)(4,6,8,7)(10,13)(11,15)(17,19)(18,24)(20,28)(21,25)(22,29)(23,30)(26,31)(27,32);(2,3)(6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,25)(20,28)(21,24)(22,29)(23,30)(26,32)(27,31);(9,11,12,15)(10,14,13,16)(17,20,23,22)(18,26,21,27)(19,28,30,29)(24,31,25,32);(1,3,5,2)(4,7,8,6)(17,21,23,18)(19,25,30,24)(20,27,22,26)(28,32,29,31);(1,4)(2,6)(3,7)(5,8)(17,22,23,20)(18,27,21,26)(19,29,30,28)(24,32,25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=gh,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, 8002113);