#8496471

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

4944 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,21)(22,24)(23,25)(27,31)(28,32)(29,30);(2,6)(3,8)(10,13)(14,15)(17,18)(19,26)(20,29)(21,30)(22,24)(23,25)(27,28)(31,32);(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,24,25,26)(20,27,30,32)(21,28,29,31);(2,6)(4,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,21)(19,27)(22,32)(23,30)(24,28)(25,29)(26,31);(1,3)(2,7)(4,6)(5,8)(17,21)(18,20)(19,28)(22,31)(23,29)(24,27)(25,30)(26,32);(1,4,5,7)(2,3,6,8)(17,22,23,19)(18,24,25,26)(20,27,30,32)(21,31,29,28)

Polycyclic presentation:

2^9|c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308955, #72309006
maximal normal subgroup of	#72308955, #72309006

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8403667);