#10519379

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

5055 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)(11,16,13,15)(17,18)(19,23)(20,30)(21,25)(22,26)(24,29)(27,32)(28,31);(1,3,5,4)(2,6,7,8)(19,27)(20,29)(23,32)(24,30);(9,11,14,13)(10,15,12,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,25)(26,32)(30,31);(1,2)(3,8)(4,6)(5,7)(9,11,14,13)(10,16,12,15)(17,20,22,29)(18,24,26,30)(19,28,27,21)(23,25,32,31);(1,2)(3,6)(4,8)(5,7)(9,12,14,10)(11,16,13,15)(17,21,22,28)(18,25,26,31)(19,29,27,20)(23,24,32,30);(1,4,5,3)(2,6,7,8)(9,13,14,11)(10,16,12,15)(17,18)(19,23)(20,24)(21,31)(22,26)(25,28)(27,32)(29,30)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10426575);