#8109958

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	#59303	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017154);