#244429

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	#2397	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

624 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)(3,11,14,7,13,16,12,15)(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)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19)(18,24)(20,29)(21,34)(22,31)(23,30)(25,38)(26,41)(27,37)(28,35)(32,43)(33,45)(36,44)(39,47)(40,48)(42,46);(2,8)(4,5)(7,11)(9,10)(12,14)(15,16)(20,32)(24,37)(25,39)(28,41)(29,43)(33,36)(40,46)(44,45);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(17,20,21,33,22,32,23,36)(18,25,26,40,27,39,35,46)(19,29,34,45,31,43,30,44)(24,38,41,48,37,47,28,42)

Polycyclic presentation:

2^9|a^2=eh,d^2=ef,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, 151625);