#6346052

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
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

1524 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253248);