#8119275

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

5697 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)(3,5)(4,7)(6,8)(10,12)(14,16)(17,18)(19,27)(20,30)(21,29)(22,23)(24,28)(25,31)(26,32);(2,3)(6,7)(11,15)(14,16)(18,23)(19,28)(25,30)(26,29);(1,2,5,3)(4,6,8,7)(9,10,13,12)(11,14,15,16)(18,23)(20,31)(24,27)(26,29);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,14,12,16)(17,19)(18,24)(20,29)(21,25)(22,28)(23,27)(26,31)(30,32);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,20,22,31)(18,25,23,30)(19,29,28,26)(21,27,32,24);(17,21)(18,26)(19,25)(20,27)(22,32)(23,29)(24,31)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312548, #72342308, #72342323
maximal normal subgroup of	#72312548, #72342308, #72342323

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026471);