#8695347

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

5622 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283347, #72312209, #72342308, …
maximal normal subgroup of	#72283347, #72312209, #72342308, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8602543);