#8417304

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	#54953	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324500);