#7798060

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	#50354	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 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)(11,16)(12,15)(13,17)(14,18)(19,23)(20,21)(22,24)(25,26,31,35)(27,32,30,34)(28,29,39,33)(36,37,38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27,31,30)(26,32,35,34)(28,36,39,38)(29,37,33,40);(11,18)(14,16)(19,24)(22,23)(25,28,31,39)(26,33,35,29)(27,36,30,38)(32,40,34,37);(9,12,13,21)(10,15,17,20)(11,19,18,24)(14,22,16,23)(25,28)(26,29)(27,36)(30,38)(31,39)(32,37)(33,35)(34,40);(1,4,5,2)(3,7,8,6)(9,10)(11,16)(12,20)(13,17)(14,18)(15,21)(19,22)(23,24)(25,29)(26,28)(27,37)(30,40)(31,33)(32,36)(34,38)(35,39);(9,11)(10,16)(12,19)(13,18)(14,17)(15,23)(20,22)(21,24)(25,30,31,27)(26,34,35,32)(28,38,39,36)(29,40,33,37)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7705256);