#7655972

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4470 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)(12,20)(15,22)(18,24)(21,23)(25,26,31,36)(27,34,39,32)(28,38,37,33)(29,30,35,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(25,27)(26,32)(28,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,31,37)(26,33,36,38)(27,35,39,29)(30,32,40,34);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,29,31,35)(26,30,36,40)(27,37,39,28)(32,38,34,33);(1,4,5,3)(2,8,6,7)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,27)(26,34)(28,35)(29,37)(30,33)(31,39)(32,36)(38,40);(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,30)(26,35)(27,38)(28,32)(29,36)(31,40)(33,39)(34,37)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=g,e^2=gh,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, 7563168);