#7898054

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

4836 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=gh,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, 7805250);