#7919683

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

4824 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)(9,10,13,15)(11,16,14,12)(17,18)(19,21)(20,28)(22,24)(23,26)(25,30)(27,29)(31,32);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,14)(15,16)(17,19)(18,23)(20,27)(21,24)(22,26)(25,29)(28,31)(30,32);(9,12,13,16)(10,14,15,11)(18,24)(19,26)(25,28)(27,32);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,23)(20,27)(21,24)(22,26)(25,29)(28,31)(30,32);(9,11)(10,12)(13,14)(15,16)(17,20,22,30)(18,25,24,28)(19,27,26,32)(21,31,23,29);(1,4,5,3)(2,8,6,7)(17,21,22,23)(18,19,24,26)(20,29,30,31)(25,32,28,27)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=gh,e^2=i,f^2=ghi,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, 7826879);