#7888604

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	#39552	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4347 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)(9,10,16,22)(11,17,13,19)(12,20,14,18)(15,21,24,23)(25,26)(27,29)(28,38)(30,33)(31,34)(32,37)(35,39)(36,40);(1,2,4,5)(3,6,7,8)(9,11,16,13)(10,17,22,19)(12,23,14,21)(15,18,24,20)(25,27)(26,32)(28,35)(29,34)(30,36)(31,37)(33,39)(38,40);(9,12,16,14)(10,18,22,20)(11,21,13,23)(15,19,24,17)(25,28,31,30)(26,33,34,38)(27,35,37,36)(29,40,32,39);(9,13,16,11)(10,19,22,17)(12,21,14,23)(15,20,24,18)(25,29,31,32)(26,27,34,37)(28,39,30,40)(33,36,38,35);(9,14,16,12)(10,20,22,18)(11,21,13,23)(15,19,24,17)(25,27)(26,32)(28,35)(29,34)(30,36)(31,37)(33,39)(38,40);(1,3,4,7)(2,6,5,8)(9,15,16,24)(10,21,22,23)(11,18,13,20)(12,19,14,17)(25,30,31,28)(26,33,34,38)(27,36,37,35)(29,40,32,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=gi,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, 7795800);