#7684610

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

5089 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7591806);