#7658153

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	#37541	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4234 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)(11,19)(12,16)(13,17)(14,18)(15,22)(20,23)(21,24)(25,26,30,34)(27,35,38,31)(28,32,29,33)(36,40,37,39);(1,2,4,5)(3,6,7,8)(25,27,30,38)(26,31,34,35)(28,36,29,37)(32,39,33,40);(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,19)(23,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,40)(34,35);(9,12,14,21)(10,16,18,24)(11,20,22,13)(15,17,19,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,13,14,20)(10,17,18,23)(11,21,22,12)(15,16,19,24)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,35,40);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,23)(14,18)(17,20)(19,22)(21,24)(25,29,30,28)(26,33,34,32)(27,37,38,36)(31,40,35,39)

Polycyclic presentation:

2^9|a^2=i,b^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, 7565349);