#7987296

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,8)(3,6,4,7)(9,10)(11,17)(12,14)(13,16)(15,18)(19,23)(20,21)(22,24)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,40)(37,39);(1,2)(3,6)(4,7)(5,8)(9,10,14,12)(11,15,18,17)(13,20,21,16)(19,23,24,22)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(10,12)(11,18)(16,20)(19,24)(25,26,30,34)(27,31,29,33)(28,32,38,36)(35,39,37,40);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,38)(37,40);(1,3,5,4)(2,6,8,7)(9,12,14,10)(11,17,18,15)(13,16,21,20)(19,22,24,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,4,5,3)(2,7,8,6)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=gi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7894492);