#7160917

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42664	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1551 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7068113);