#244162

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	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=efh,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, 151358);