#7816047

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

5093 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)(3,7)(4,6)(5,8)(10,13)(14,16)(17,18,23,20)(19,27,28,24)(21,25,31,29)(22,26,32,30);(9,10,12,13)(11,14,15,16)(17,19,23,28)(18,24,20,27)(21,22,31,32)(25,30,29,26);(1,2,4,6)(3,5,8,7)(19,28)(22,32)(24,27)(26,30);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,14,13,16)(17,20,23,18)(19,27,28,24)(21,25,31,29)(22,30,32,26);(1,3)(2,5)(4,8)(6,7)(17,21)(18,25)(19,22)(20,29)(23,31)(24,30)(26,27)(28,32);(17,22)(18,26)(19,21)(20,30)(23,32)(24,29)(25,27)(28,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,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, 7723243);