#6345992

order	512
nilpotency class	3
isoclinism	isoclinic class of #62295
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61447	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1507 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=chi,d^a=dh,f^a=fh,c^b=ci,f^b=fh,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253188);