#8746620

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

5972 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,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.

Relationships

maximal subgroup of	#72312909, #72342307, #72342319, …
maximal normal subgroup of	#72312909, #72342307, #72342319, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653816);