#7972300

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

4232 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=h,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, 7879496);