#7157048

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45387	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1894 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)(20,31)(21,26)(22,28)(23,24)(25,33)(29,40)(30,41)(32,44)(34,37)(35,38)(36,42)(39,45)(43,48)(46,47);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15)(17,19,22,23)(18,24,28,27)(20,29,34,33)(21,35,36,30)(25,37,40,31)(26,41,42,38)(32,45,46,43)(39,44,48,47);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,33,22,34,19,29)(18,25,27,31,28,40,24,37)(21,32,35,45,36,46,30,43)(26,39,41,44,42,48,38,47);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(17,21,22,36)(18,26,28,42)(19,30,23,35)(20,32,34,46)(24,38,27,41)(25,39,40,48)(29,43,33,45)(31,44,37,47);(9,11,12,15)(10,14,13,16)(17,19,22,23)(18,27,28,24)(20,33,34,29)(21,35,36,30)(25,37,40,31)(26,38,42,41)(32,43,46,45)(39,44,48,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,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=eh,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, 7064244);