#7783952

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	#40148	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 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)(25,26)(27,32)(28,37)(29,34)(30,35)(31,38)(33,39)(36,40);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(10,15)(13,16)(17,23)(21,24)(25,27,30,38)(26,31,35,32)(28,36,39,34)(29,33,40,37);(1,2,5,4)(3,6,8,7)(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,21,23,24)(25,27)(26,32)(28,34)(29,37)(30,38)(31,35)(33,40)(36,39);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,24)(18,20)(25,28)(26,33)(27,36)(29,32)(30,39)(31,40)(34,38)(35,37);(1,4,5,2)(3,7,8,6)(9,13,14,16)(10,18,15,11)(12,21,22,24)(17,20,23,19)(25,29)(26,34)(27,37)(28,32)(30,40)(31,39)(33,38)(35,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gi,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, 7691148);