#7917733

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

4931 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gi,e^2=i,f^2=gi,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, 7824929);