#8097732

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	C₂×#550	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4621 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004928);