#8438171

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

5028 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,30)(21,26)(22,27)(23,28)(25,29)(31,32);(1,3,5,4)(2,6,7,8)(9,10,12,14)(11,13,16,15)(17,19,23,22)(18,24,28,27)(20,31,29,21)(25,26,30,32);(9,10)(11,13)(12,14)(15,16)(17,20)(18,25)(19,21)(22,31)(23,29)(24,32)(26,27)(28,30);(2,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,21,23,31)(18,26,28,32)(19,29,22,20)(24,25,27,30);(1,4,5,3)(2,8,7,6)(17,22,23,19)(18,27,28,24)(20,31,29,21)(25,26,30,32);(1,2)(3,6)(4,8)(5,7)(17,18)(19,24)(20,25)(21,32)(22,27)(23,28)(26,31)(29,30)

Polycyclic presentation:

2^9|b^2=gh,d^2=h,e^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, 8345367);