#7697266

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

4768 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7604462);