#7671218

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	#59313	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5126 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)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,27,28,24)(21,25,31,29)(22,30,32,26);(9,11,13,16)(10,14,12,15)(17,19)(18,24)(20,27)(21,22)(23,28)(25,26)(29,30)(31,32);(2,7)(3,8)(18,20)(19,28)(21,31)(26,30);(1,3,6,8)(2,4,7,5)(17,20,23,18)(19,27,28,24)(21,29,31,25)(22,30,32,26);(1,4,6,5)(2,3,7,8)(9,10,13,12)(11,14,16,15)(17,21)(18,25)(19,22)(20,29)(23,31)(24,26)(27,30)(28,32);(1,5,6,4)(2,8,7,3)(9,12,13,10)(11,15,16,14)(17,22)(18,26)(19,21)(20,30)(23,32)(24,25)(27,29)(28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=h,e^2=gh,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, 7578414);