#7983184

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	#36231	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4369 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,f^2=hi,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, 7890380);