#7773958

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

5052 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7681154);