#7715818

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	#45135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 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)(12,14)(13,21)(15,17)(18,22)(19,23)(20,24)(25,26,31,29)(27,34,37,32)(28,38,30,33)(35,39,36,40);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,23,21,24)(25,27)(26,32)(28,35)(29,34)(30,36)(31,37)(33,39)(38,40);(1,2,5,4)(3,6,8,7)(10,17)(11,12)(13,22)(23,24)(25,28,31,30)(26,33,29,38)(27,35,37,36)(32,39,34,40);(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,23,21,24)(25,29,31,26)(27,34,37,32)(28,33,30,38)(35,39,36,40);(9,13)(10,18)(11,19)(12,20)(14,23)(15,22)(16,24)(17,21)(25,27)(26,32)(28,35)(29,34)(30,36)(31,37)(33,39)(38,40);(1,4,5,2)(3,7,8,6)(9,14)(10,11)(12,17)(13,23)(15,16)(18,19)(20,21)(22,24)(25,30,31,28)(26,33,29,38)(27,36,37,35)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7623014);