#8751502

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

5044 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)(3,5)(4,7)(6,8)(9,10,14,12)(11,15,13,16)(17,18)(19,27)(20,26)(21,24)(22,28)(23,25)(29,32)(30,31);(2,3)(6,7)(9,11,14,13)(10,15,12,16)(17,19,23,21)(18,24,25,27)(20,31,28,29)(22,30,26,32);(1,2,5,3)(4,6,8,7)(9,11,14,13)(10,16,12,15)(18,25)(20,28)(24,27)(29,31);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,13,16)(17,20)(18,26)(19,29)(21,31)(22,25)(23,28)(24,30)(27,32);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,15,12,16)(17,21,23,19)(18,27,25,24)(20,31,28,29)(22,30,26,32);(17,22,23,26)(18,28,25,20)(19,30,21,32)(24,29,27,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8658698);