#237182

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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1108 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,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, 144378);