#7199740

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1241 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,21)(14,17)(18,20)(19,23)(22,24)(25,26,31,29,32,36,30,35)(27,34,38,37,28,39,40,33);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,27,32,28)(26,33,36,37)(29,39,35,34)(30,38,31,40);(1,4,5,3)(2,6,7,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,28,32,27)(26,34,36,39)(29,37,35,33)(30,38,31,40);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29,30,26,32,35,31,36)(27,37,40,34,28,33,38,39);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,30,32,31)(26,35,36,29)(27,38,28,40)(33,34,37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=fhi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106936);