#7184442

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1421 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,23,36,22,27,21,28)(19,25,35,24,30,41,31,39)(20,26,37,47,34,42,33,43)(29,40,46,38,44,48,32,45);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19,22,30)(18,24,27,39)(20,32,34,46)(21,35,23,31)(25,28,41,36)(26,40,42,48)(29,37,44,33)(38,47,45,43);(2,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,29)(21,31)(22,30)(23,35)(24,36)(26,40)(27,41)(28,39)(32,33)(34,44)(37,46)(38,47)(42,48)(43,45);(10,13)(14,16)(17,20,21,33,22,34,23,37)(18,26,28,43,27,42,36,47)(19,29,31,32,30,44,35,46)(24,38,25,40,39,45,41,48);(9,11,12,15)(10,14,13,16)(18,27)(19,30)(26,42)(28,36)(29,44)(31,35)(32,46)(43,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,d^2=f,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, 7091638);