#7783249

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	#47843	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

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, 7690445);