#7647075

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	#41843	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4237 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)(9,10,15,19)(11,18,14,16)(12,13,23,17)(20,24,22,21)(25,26,31,35)(27,34,30,32)(28,38,33,29)(36,40,39,37);(1,3,5,8)(2,6,4,7)(9,11,15,14)(10,16,19,18)(12,20,23,22)(13,21,17,24)(25,27,31,30)(26,32,35,34)(28,39,33,36)(29,37,38,40);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,20)(13,19)(14,22)(15,23)(16,24)(18,21)(25,28,31,33)(26,29,35,38)(27,36,30,39)(32,40,34,37);(1,2,5,4)(3,6,8,7)(9,13,15,17)(10,12,19,23)(11,21,14,24)(16,20,18,22)(25,29)(26,33)(27,37)(28,35)(30,40)(31,38)(32,36)(34,39);(9,12)(10,13)(11,20)(14,22)(15,23)(16,21)(17,19)(18,24)(25,27,31,30)(26,34,35,32)(28,36,33,39)(29,37,38,40);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,18,19,16)(12,22,23,20)(13,24,17,21)(25,30,31,27)(26,32,35,34)(28,39,33,36)(29,40,38,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7554271);