#7656568

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

4745 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(9,10,13,12)(11,15,16,14)(17,18,23,21)(19,26,22,24)(20,28,30,25)(27,31,29,32);(1,3,5,6)(2,4,7,8)(9,11,13,16)(10,14,12,15)(17,19,23,22)(18,24,21,26)(20,27,30,29)(25,31,28,32);(1,3,5,6)(2,4,7,8)(9,10,13,12)(11,14,16,15)(17,18,23,21)(19,26,22,24)(20,25,30,28)(27,32,29,31);(9,12,13,10)(11,15,16,14)(17,20,23,30)(18,25,21,28)(19,27,22,29)(24,31,26,32);(1,2)(3,4)(5,7)(6,8)(17,21,23,18)(19,26,22,24)(20,28,30,25)(27,32,29,31);(1,4)(2,6)(3,7)(5,8)(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=g,c^2=g,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7563764);