#265413

order	512
nilpotency class	3
isoclinism	isoclinic class of #265399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#18229	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

914 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)(3,9)(4,7)(5,8)(6,11)(10,13)(12,15)(14,16)(17,18)(19,25)(20,33)(21,36)(22,29)(23,28)(24,31)(26,37)(27,34)(30,45)(32,40)(35,39)(38,44)(41,46)(42,47)(43,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,14,15,13)(17,19,22,31,23,20,34,32)(18,24,27,25,28,40,29,33)(21,30,37,46,38,35,48,47)(26,39,36,41,43,45,44,42);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(17,20)(18,25)(19,23)(21,30)(22,32)(24,29)(26,42)(27,40)(28,33)(31,34)(35,38)(36,39)(37,46)(41,43)(44,45)(47,48);(2,8)(4,12)(6,9)(10,16)(17,21,34,48,23,38,22,37)(18,26,29,44,28,43,27,36)(19,30,32,47,20,35,31,46)(24,39,33,42,40,45,25,41)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172609);