#7049525

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

1333 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)(17,18,22,24)(19,29,30,43)(20,26,33,39)(21,28,23,27)(25,36,38,32)(31,45,34,48)(35,44,37,42)(40,46,41,47);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,16,15,14)(18,24)(19,30)(20,33)(27,28)(32,36)(35,37)(40,41)(45,48);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,36,22,30,23,32)(18,25,27,43,24,38,28,29)(20,34,37,47,33,31,35,46)(26,41,44,45,39,40,42,48);(3,8)(6,7)(10,13)(14,16)(17,20,22,33)(18,26,24,39)(19,31,30,34)(21,37,23,35)(25,40,38,41)(27,44,28,42)(29,45,43,48)(32,47,36,46);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,24,28)(19,32,30,36)(20,35,33,37)(25,29,38,43)(26,42,39,44)(31,46,34,47)(40,48,41,45)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=fg,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=dh,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, 6956721);