#7969575

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

4974 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7876771);