#8505379

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

5120 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)(9,10,14,12)(11,15,13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32);(1,3,6,5)(2,7,4,8)(9,11,14,13)(10,15,12,16)(17,19,22,25)(18,23,26,21)(20,28,30,27)(24,29,32,31);(9,11,14,13)(10,16,12,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,30)(31,32);(1,2,6,4)(3,7,5,8)(9,12,14,10)(11,15,13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32);(1,4,6,2)(3,7,5,8)(9,13,14,11)(10,15,12,16)(17,20,22,30)(18,24,26,32)(19,27,25,28)(21,29,23,31);(1,5,6,3)(2,7,4,8)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,28,32,27)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412575);