#244422

order	512
nilpotency class	3
isoclinism	isoclinic class of #232749
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	#10012	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151618);