#7685070

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

4898 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,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, 7592266);