#238987

order	512
nilpotency class	3
isoclinism	isoclinic class of #10530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	#60907	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 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:

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

Polycyclic presentation:

2^9|a^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=gh,c^b=cg,e^b=eh,f^b=fh,g^b=ghi,f^c=fhi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146183);