#7049777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,4)(3,7,8,6)(10,13)(11,15)(17,18)(19,27)(20,24)(21,34)(22,25)(23,31)(26,29)(28,39)(30,40)(32,38)(33,45)(35,42)(36,44)(37,43)(41,47)(46,48);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,16,15,14)(19,28)(23,35)(27,39)(30,41)(31,42)(36,46)(40,47)(44,48);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,29,42,22,28,21,31)(18,23,34,39,25,35,26,27)(20,30,43,48,32,41,33,44)(24,36,45,47,38,46,37,40);(3,8)(6,7)(10,13)(14,16)(17,20,22,32)(18,24,25,38)(19,30,28,41)(21,33,29,43)(23,36,35,46)(26,37,34,45)(27,40,39,47)(31,44,42,48);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(18,25)(19,28)(20,32)(26,34)(31,42)(33,43)(36,46)(40,47)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=fgi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,e^d=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, 6956973);