#7658180

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	#41927	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4222 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)(9,10)(11,20)(12,17)(13,16)(14,18)(15,19)(21,22)(23,24)(25,26,31,35)(27,34,30,32)(28,38,33,29)(36,40,39,37);(1,2,5,6)(3,7,4,8)(9,11,15,13)(10,16,19,20)(12,21,18,23)(14,22,17,24)(25,27,31,30)(26,32,35,34)(28,39,33,36)(29,37,38,40);(1,3,5,4)(2,7,6,8)(9,11,15,13)(10,16,19,20)(12,21,18,23)(14,22,17,24)(25,28,31,33)(26,29,35,38)(27,36,30,39)(32,40,34,37);(9,12,15,18)(10,17,19,14)(11,21,13,23)(16,24,20,22)(25,29)(26,33)(27,37)(28,35)(30,40)(31,38)(32,36)(34,39);(9,13,15,11)(10,16,19,20)(12,23,18,21)(14,22,17,24)(25,27,31,30)(26,34,35,32)(28,36,33,39)(29,37,38,40);(1,4,5,3)(2,8,6,7)(9,14,15,17)(10,18,19,12)(11,22,13,24)(16,23,20,21)(25,30,31,27)(26,32,35,34)(28,39,33,36)(29,40,38,37)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7565376);