#285438

order	512
nilpotency class	3
isoclinism	isoclinic class of #285371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

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, 192634);