#7052003

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,d^2=f,e^2=f,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=dh,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, 6959199);