#8057946

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

4415 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)(3,4)(5,8)(6,7)(10,13)(14,16)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28);(1,3)(2,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(18,23)(20,27)(25,29)(28,32);(1,2,7,6)(3,5,8,4)(9,11,12,15)(10,14,13,16)(18,23)(19,24)(25,29)(26,31);(1,4,7,5)(2,8,6,3)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32);(1,5,7,4)(2,8,6,3)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32);(1,6,7,2)(3,5,8,4)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

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, 7965142);