#8461705

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

5018 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=g,e^2=i,f^2=hi,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, 8368901);