#7837518

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

4748 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7744714);