#244434

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	#10032	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 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)(3,11,12,7,14,15,13,16)(17,18,23,35,22,27,21,26)(19,28,34,24,31,41,30,37)(20,25,36,46,32,39,33,40)(29,42,45,38,43,48,44,47);(1,3,6,14)(2,7,10,16)(4,12,5,13)(8,15,9,11)(17,19,22,31)(18,24,27,37)(20,29,32,43)(21,34,23,30)(25,38,39,47)(26,41,35,28)(33,45,36,44)(40,48,46,42);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(20,32)(24,37)(25,39)(28,41)(29,43)(33,36)(40,46)(44,45);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,23,36,22,32,21,33)(18,25,35,46,27,39,26,40)(19,29,30,44,31,43,34,45)(24,38,28,42,37,47,41,48)

Polycyclic presentation:

2^9|a^2=eh,b^2=h,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, 151630);