#7684846

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

5353 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)(3,4)(5,7)(6,8)(10,14)(15,16)(17,18)(19,21)(20,25)(22,26)(23,24)(27,29)(28,30)(31,32);(9,10,13,14)(11,15,12,16)(17,19,23,20)(18,21,24,25)(22,27,31,28)(26,29,32,30);(1,3,6,5)(2,7,8,4)(18,24)(19,20)(26,32)(27,28);(1,4,6,7)(2,3,8,5)(9,11,13,12)(10,15,14,16)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,30,32,29);(1,5,6,3)(2,7,8,4)(9,12,13,11)(10,16,14,15)(17,21,23,25)(18,19,24,20)(22,30,31,29)(26,28,32,27);(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=gh,e^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, 7592042);