#8081038

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

5048 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)(3,5)(4,6)(7,8)(10,13)(14,16)(17,18,22,23)(19,25,26,20)(21,24,29,31)(27,30,32,28);(9,10,12,13)(11,14,15,16)(17,19,22,26)(18,20,23,25)(21,27,29,32)(24,28,31,30);(1,2,4,6)(3,5,7,8)(9,11,12,15)(10,14,13,16)(18,23)(19,26)(21,29)(28,30);(3,7)(5,8)(17,20)(18,19)(21,30)(22,25)(23,26)(24,32)(27,31)(28,29);(1,3)(2,5)(4,7)(6,8)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,31)(30,32);(1,2)(3,5)(4,6)(7,8)(17,21)(18,24)(19,27)(20,28)(22,29)(23,31)(25,30)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7988234);