#232786

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#60951	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1203 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=di,e^b=eh,g^b=gh,e^c=ei,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, 139982);