#7942164

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

4881 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)(12,14)(13,16)(17,18,23,21)(19,26,22,24)(20,28,30,25)(27,31,29,32);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,22)(18,24,21,26)(20,27,30,29)(25,31,28,32);(10,14)(13,15)(17,18,23,21)(19,26,22,24)(20,25,30,28)(27,32,29,31);(9,10)(11,13)(12,14)(15,16)(17,20,23,30)(18,25,21,28)(19,27,22,29)(24,31,26,32);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,21,23,18)(19,26,22,24)(20,28,30,25)(27,32,29,31);(1,4,5,3)(2,7,8,6)(17,22,23,19)(18,24,21,26)(20,29,30,27)(25,31,28,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=i,e^2=gh,f^2=gh,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, 7849360);