#7847404

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

5036 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=g,e^2=i,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, 7754600);