#244158

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
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	#532	C₂³
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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=ef,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,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, 151354);