#8109961

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4704 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)(9,10,13,11)(12,15,16,14)(17,18,22,25)(19,21,27,23)(20,28,30,24)(26,31,32,29);(2,3)(6,7)(10,11)(14,15)(17,19,22,27)(18,23,25,21)(20,26,30,32)(24,31,28,29);(1,2,5,3)(4,6,8,7)(9,11,13,10)(12,15,16,14)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32);(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,29)(30,32);(1,3,5,2)(4,7,8,6)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,28,32);(1,4)(2,6)(3,7)(5,8)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=gh,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, 8017157);