#7748724

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

5082 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:

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

Polycyclic presentation:

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

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, 7655920);