#8456817

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

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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=h,e^2=hi,f^2=g,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, 8364013);