#244186

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
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	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

590 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:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151382);