#240024

order	512
nilpotency class	3
isoclinism	isoclinic class of #240017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

793 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:

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

Polycyclic presentation:

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

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, 147220);