#8464216

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

4895 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)(3,4)(5,7)(6,8)(10,15)(11,12)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,13)(14,15)(19,30)(20,31)(24,29)(25,32);(9,11,14,12)(10,16,15,13)(17,19)(18,24)(20,27)(21,31)(22,32)(23,30)(25,26)(28,29);(2,7)(4,6)(9,12,14,11)(10,16,15,13)(17,20,23,31)(18,25,28,32)(19,27,30,21)(22,24,26,29);(1,3)(2,6)(4,7)(5,8)(17,21,23,27)(18,26,28,22)(19,31,30,20)(24,25,29,32);(1,4,5,6)(2,3,7,8)(9,13,14,16)(10,12,15,11)(17,22,23,26)(18,27,28,21)(19,32,30,25)(20,29,31,24)

Polycyclic presentation:

2^9|c^2=hi,d^2=h,e^2=i,f^2=gi,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, 8371412);