#8599386

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	#37660	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5211 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506582);