#7972332

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

4193 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)(19,24)(20,22)(21,23)(25,26,31,30)(27,35,36,32)(28,39,38,33)(29,34,37,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,27)(26,32)(28,37)(29,38)(30,35)(31,36)(33,40)(34,39);(1,3,5,4)(2,6,8,7)(10,12)(11,18)(16,20)(19,21)(27,36)(29,37)(32,35)(34,40);(1,3,5,4)(2,6,8,7)(9,12,14,10)(11,17,18,15)(13,20,22,16)(19,24,21,23)(25,28,31,38)(26,33,30,39)(27,37,36,29)(32,40,35,34);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,29,31,37)(26,34,30,40)(27,38,36,28)(32,39,35,33);(1,4,5,3)(2,7,8,6)(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(25,30,31,26)(27,32,36,35)(28,39,38,33)(29,34,37,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,e^2=gh,f^2=g,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, 7879528);