#7652790

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

4970 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)(10,13)(11,15)(17,18)(19,25)(20,24)(21,23)(22,27)(26,32)(28,30)(29,31);(1,2)(3,6)(4,7)(5,8)(9,10,12,13)(11,14,15,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,29)(31,32);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,23)(22,25)(24,31)(30,32);(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31);(17,20,21,29)(18,24,23,31)(19,26,27,28)(22,30,25,32);(1,4,5,3)(2,7,8,6)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=i,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7559986);