#7897861

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

4534 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,e^2=i,f^2=gi,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, 7805057);