#244133

order	512
nilpotency class	3
isoclinism	isoclinic class of #232760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	#10032	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,d^2=ef,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151329);