#237132

order	512
nilpotency class	3
isoclinism	isoclinic class of #237115
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

1208 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:

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

Polycyclic presentation:

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

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, 144328);