#7901377

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	#39889	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4580 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7808573);