#8109986

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	#36727	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4866 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,e^2=ghi,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, 8017182);