#7672297

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	#59303	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5094 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579493);