#8933765

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	#35390	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5626 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=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, 8840961);