#7689147

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

5313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596343);