#242378

order	512
nilpotency class	3
isoclinism	isoclinic class of #242377
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

638 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=dh,g^b=gi,e^c=eh,f^c=fi,g^c=ghi

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, 149574);