#8004027

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

4914 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7911223);