#8199230

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	#37648	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4840 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8106426);