#7156976

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45388	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1861 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)(11,15)(14,16)(17,18)(19,27)(20,25)(21,23)(22,34)(24,31)(26,29)(28,40)(30,39)(32,36)(33,45)(35,42)(37,44)(38,43)(41,47)(46,48);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15)(18,23)(19,28)(20,32)(26,34)(31,42)(33,43)(37,46)(39,47);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,29,42,21,28,22,31)(18,24,34,40,23,35,26,27)(20,30,43,48,32,41,33,44)(25,37,45,47,36,46,38,39);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(17,20,21,32)(18,25,23,36)(19,30,28,41)(22,33,29,43)(24,37,35,46)(26,38,34,45)(27,39,40,47)(31,44,42,48);(9,11,12,15)(10,14,13,16)(18,23)(20,32)(24,35)(26,34)(27,40)(30,41)(33,43)(44,48)

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,e^2=hi,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^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7064172);