#7684567

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

4743 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)(3,7)(4,8)(5,6)(10,13)(14,16)(17,18)(19,22)(20,25)(21,26)(23,24)(27,31)(28,29)(30,32);(9,10,12,13)(11,14,15,16)(17,19,23,20)(18,22,24,25)(21,27,30,28)(26,31,32,29);(2,6)(3,8)(18,24)(19,20)(26,32)(27,28);(1,3,5,8)(2,7,6,4)(9,11,12,15)(10,14,13,16)(17,20,23,19)(18,25,24,22)(21,28,30,27)(26,29,32,31);(1,4,5,7)(2,8,6,3)(17,21)(18,26)(19,27)(20,28)(22,31)(23,30)(24,32)(25,29);(17,22,23,25)(18,19,24,20)(21,29,30,31)(26,28,32,27)

Polycyclic presentation:

2^9|b^2=gi,d^2=gh,e^2=hi,f^2=i,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, 7591763);