#7888968

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

4817 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=gh,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7796164);