#237414

order	512
nilpotency class	3
isoclinism	isoclinic class of #237395
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

1148 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)(3,11,14,16)(4,8,5,9)(7,12,15,13)(17,18,23,28)(19,29,33,39)(20,25,35,42)(21,27,22,26)(24,32,40,31)(30,41,45,48)(34,38,46,47)(36,44,37,43);(1,3,6,14)(2,7,10,15)(4,12,5,13)(8,11,9,16)(17,19,23,33)(18,24,28,40)(20,34,35,46)(21,32,22,31)(25,41,42,48)(26,39,27,29)(30,36,45,37)(38,43,47,44);(1,3)(2,7)(4,13)(5,12)(6,14)(8,16)(9,11)(10,15)(17,20)(18,25)(19,30)(21,37)(22,36)(23,35)(24,38)(26,44)(27,43)(28,42)(29,41)(31,34)(32,46)(33,45)(39,48)(40,47);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,15,16)(20,35)(25,42)(30,45)(34,46)(36,37)(38,47)(41,48)(43,44)

Polycyclic presentation:

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

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, 144610);