#6344429

order	512
nilpotency class	3
isoclinism	isoclinic class of #61559
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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	C₂³×C₄⋊C₄
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1649 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,21,25,22,26,32,38)(19,23,29,35,28,34,41,45)(20,24,31,37,30,36,42,46)(27,33,40,44,39,43,47,48);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(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);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,33)(25,37)(26,36)(28,39)(29,40)(32,42)(34,43)(35,44)(38,46)(41,47)(45,48);(20,30)(24,36)(27,39)(31,42)(33,43)(37,46)(40,47)(44,48);(19,28)(23,34)(27,39)(29,41)(33,43)(35,45)(40,47)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251625);