#7188972

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=i,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7096168);