#7197673

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=h,d^2=fgi,e^2=h,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,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, 7104869);