#9434291

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

5394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219217, #72219279, #72219285
maximal quotient of	#72219214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9341487);