#6346266

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	#61446	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2965 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1492 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)(19,29)(20,26)(21,28)(22,27)(23,25)(24,31)(30,43)(32,42)(33,39)(34,41)(35,40)(36,37)(38,46)(44,48)(45,47);(2,7)(3,9)(8,14)(10,15)(17,19,23,32)(18,24,25,37)(20,30,33,44)(21,31,22,36)(26,38,39,47)(27,29,28,42)(34,45,35,46)(40,48,41,43);(1,3,6,7,5,9,13,2)(4,10,12,14,11,15,16,8)(18,25)(20,33)(24,37)(27,28)(29,42)(30,44)(34,35)(45,46);(1,4)(2,8)(3,10)(5,11)(6,12)(7,14)(9,15)(13,16)(17,20)(18,26)(19,30)(21,35)(22,34)(23,33)(24,38)(25,39)(27,41)(28,40)(29,43)(31,46)(32,44)(36,45)(37,47)(42,48);(4,11)(8,14)(10,15)(12,16)(17,21,23,22)(18,27,25,28)(19,31,32,36)(20,34,33,35)(24,29,37,42)(26,40,39,41)(30,45,44,46)(38,48,47,43)

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,c^a=chi,e^a=eh,f^a=fh,c^b=ci,f^b=fh,d^c=dh,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, 6253462);