#240990

order	512
nilpotency class	3
isoclinism	isoclinic class of #240979
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	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

819 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148186);