#237078

order	512
nilpotency class	3
isoclinism	isoclinic class of #237043
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1136 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,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=di,e^b=eh,f^b=fhi,g^b=gh,e^c=ei,f^c=fhi,g^c=gh

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, 144274);