#6346040

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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	#3475	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2922 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1527 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:

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

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=ch,e^a=eh,f^a=fh,f^b=fh,d^c=dh,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253236);