#8677372

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

4962 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8584568);