#7683440

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

4705 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)(17,18,23,27)(19,28,29,24)(20,21,25,32)(22,30,31,26);(1,3,5,8)(2,6,4,7)(9,11,14,12)(10,15,16,13)(17,19)(18,24)(20,30)(21,22)(23,29)(25,26)(27,28)(31,32);(9,12,14,11)(10,15,16,13)(19,29)(22,31)(24,28)(26,30);(9,13,14,15)(10,12,16,11)(17,20)(18,21)(19,30)(22,24)(23,25)(26,29)(27,32)(28,31);(1,2,5,4)(3,6,8,7)(17,21)(18,25)(19,31)(20,27)(22,29)(23,32)(24,30)(26,28);(1,4,5,2)(3,7,8,6)(17,22,23,31)(18,26,27,30)(19,21,29,32)(20,24,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590636);