#7711768

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	#45972	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4382 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)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,31,33)(27,34,29,32)(28,30,37,40)(35,39,38,36);(1,3,5,8)(2,6,4,7)(25,27,31,29)(26,32,33,34)(28,35,37,38)(30,36,40,39);(10,14)(15,17)(16,21)(22,23)(26,33)(27,29)(28,37)(36,39);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28)(26,30)(27,35)(29,38)(31,37)(32,36)(33,40)(34,39);(1,2,5,4)(3,6,8,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,27)(26,34,33,32)(28,38,37,35)(30,39,40,36);(1,4,5,2)(3,7,8,6)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30)(26,28)(27,36)(29,39)(31,40)(32,35)(33,37)(34,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618964);