#317216

order	512
nilpotency class	3
isoclinism	isoclinic class of #317195
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

509 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)(3,10)(4,8)(5,7)(6,11)(9,13)(12,14)(15,16)(17,18,24,28)(19,29,33,39)(20,34,38,43)(21,23,27,22)(25,32,41,31)(26,36,44,37)(30,45,46,40)(35,42,47,48);(1,3)(2,6)(4,11)(5,12)(7,15)(8,10)(9,14)(13,16)(17,19,22,31,24,33,23,32)(18,25,27,29,28,41,21,39)(20,35,37,30,38,47,36,46)(26,42,34,40,44,48,43,45);(2,7)(3,11)(5,13)(6,14)(8,9)(10,15)(17,20)(18,26)(19,30)(21,34)(22,36)(23,37)(24,38)(25,40)(27,43)(28,44)(29,42)(31,35)(32,47)(33,46)(39,48)(41,45);(2,8)(3,11)(7,9)(12,16)(17,21,22,18,24,27,23,28)(19,29,31,41,33,39,32,25)(20,34,36,26,38,43,37,44)(30,42,35,45,46,48,47,40)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224412);