#237184

order	512
nilpotency class	3
isoclinism	isoclinic class of #237155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1151 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)(3,11,14,15)(4,7,5,9)(8,13,16,12)(17,18)(19,24)(20,32)(21,26)(22,35)(23,27)(25,33)(28,43)(29,38)(30,42)(31,39)(34,40)(36,41)(37,44)(45,47)(46,48);(2,7)(3,12)(4,5)(9,10)(11,15)(13,14)(17,19,23,31)(18,24,27,39)(20,28,34,45)(21,30,22,29)(25,37,41,48)(26,42,35,38)(32,47,40,43)(33,46,36,44);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,20)(18,25)(19,28)(21,33)(22,36)(23,34)(24,37)(26,40)(27,41)(29,44)(30,46)(31,45)(32,35)(38,47)(39,48)(42,43);(1,4,6,5)(2,7,10,9)(3,13,14,12)(8,11,16,15)(17,21,23,22)(18,26,27,35)(19,29,31,30)(20,33,34,36)(24,38,39,42)(25,40,41,32)(28,44,45,46)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=dhi,e^b=eh,f^b=fhi,g^b=gh,d^c=dh,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, 144380);