#8934500

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
prime radical	#52004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5104 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318261, #72318304, #72319289
maximal normal subgroup of	#72318261, #72318304, #72319289

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841696);