#272184

order	512
nilpotency class	3
isoclinism	isoclinic class of #272155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

809 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,13)(10,14)(17,18,21,26,23,28,22,27)(19,24,30,39,31,40,37,44)(20,32,34,43,36,42,35,25)(29,41,33,47,45,48,46,38);(1,3,7,5)(2,8,11,10)(4,13,6,14)(9,15,16,12)(17,19)(18,24)(20,33)(21,30)(22,37)(23,31)(25,41)(26,39)(27,44)(28,40)(29,35)(32,47)(34,45)(36,46)(38,42)(43,48);(1,4,7,6)(2,9,11,16)(3,13,5,14)(8,15,10,12)(17,20,23,36)(18,25,28,43)(19,29,31,45)(21,35,22,34)(24,38,40,47)(26,42,27,32)(30,46,37,33)(39,48,44,41);(1,5,7,3)(2,10,11,8)(4,14,6,13)(9,12,16,15)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179380);