#7646796

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

4575 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)(10,13)(14,16)(18,20)(19,27)(25,26)(29,30);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,28,27,24)(21,25,22,26)(29,32,30,31);(1,3,5,4)(2,6,7,8)(17,19,23,27)(18,24,20,28)(21,29,22,30)(25,31,26,32);(2,7)(6,8)(17,20,23,18)(19,28,27,24)(21,26,22,25)(29,32,30,31);(1,2)(3,6)(4,8)(5,7)(17,21,23,22)(18,25,20,26)(19,29,27,30)(24,31,28,32);(1,4,5,3)(2,8,7,6)(9,11,12,15)(10,14,13,16)(17,22,23,21)(18,26,20,25)(19,30,27,29)(24,32,28,31)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7553992);