#280912

order	512
nilpotency class	3
isoclinism	isoclinic class of #280883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

465 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=ei,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gi,d^c=di,f^c=fh,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, 188108);