#8141438

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

5128 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

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, 8048634);