#7642621

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

4877 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)(3,8)(4,6)(5,7)(10,13)(14,16)(18,23)(21,31)(24,32)(25,28);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,24,29,32)(20,25,30,28)(21,26,31,27);(2,7)(6,8)(18,23)(20,30)(24,32)(26,27);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,19,22,29)(18,24,23,32)(20,27,30,26)(21,28,31,25);(1,2)(3,6)(4,8)(5,7)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32);(1,4,5,3)(2,8,7,6)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549817);