#7904549

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

5152 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=ghi,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7811745);