#6345665

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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	#60942	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,e^a=ehi,f^a=fh,e^b=ei,f^b=fh,d^c=di,e^c=eh

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, 6252861);