#8600098

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	#36212	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5584 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=h,e^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, 8507294);