#7684909

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

4690 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=i,e^2=hi,f^2=gi,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, 7592105);