#243883

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	C₂×#399
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=h,d^2=h,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, 151079);