#242388

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

581 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=h,d^2=hi,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, 149584);