#6524578

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1437 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)(3,6)(4,8)(5,7)(17,18,23,35,22,26,21,27)(19,28,34,24,31,41,30,38)(20,25,36,46,32,40,33,39)(29,42,45,37,43,48,44,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,26,38)(20,29,32,43)(21,34,23,30)(25,37,40,47)(27,41,35,28)(33,45,36,44)(39,48,46,42);(2,7)(6,8)(10,13)(11,15);(9,11,12,15)(10,14,13,16)(17,20,22,32)(18,25,26,40)(19,29,31,43)(21,33,23,36)(24,37,38,47)(27,39,35,46)(28,42,41,48)(30,44,34,45);(1,4,5,3)(2,8,7,6)(10,13)(14,16)(18,26)(20,32)(24,38)(27,35)(28,41)(29,43)(33,36)(44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6431774);