#7883003

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	#45094	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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)(9,10,16,22)(11,17,13,19)(12,20,14,18)(15,21,24,23)(25,26)(27,37)(28,29)(30,34)(31,35)(32,36)(33,40)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11,16,13)(10,17,22,19)(12,23,14,21)(15,18,24,20)(25,27)(26,33)(28,35)(29,39)(30,38)(31,34)(32,40)(36,37);(9,12,16,14)(10,18,22,20)(11,21,13,23)(15,19,24,17)(25,28,32,30)(26,29,36,34)(27,38,40,35)(31,37,39,33);(1,3,5,4)(2,6,8,7)(9,13,16,11)(10,19,22,17)(12,21,14,23)(15,20,24,18)(25,29,32,34)(26,28,36,30)(27,39,40,31)(33,35,37,38);(1,4,5,3)(2,7,8,6)(9,14,16,12)(10,20,22,18)(11,21,13,23)(15,19,24,17)(25,30,32,28)(26,34,36,29)(27,38,40,35)(31,37,39,33);(9,15,16,24)(10,21,22,23)(11,18,13,20)(12,19,14,17)(25,31,32,39)(26,35,36,38)(27,34,40,29)(28,37,30,33)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=gh,e^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, 7790199);