#8112505

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

5384 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300462
maximal normal subgroup of	#72300462
maximal quotient of	#72300499, #72300914

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019701);