#7883683

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	#39525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4439 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7790879);