#238842

order	512
nilpotency class	3
isoclinism	isoclinic class of #238834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60948	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

832 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)(3,7)(4,8)(5,9)(6,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,28,31,40)(20,32,35,25)(21,26,22,37)(24,38,41,30)(29,46,33,45)(34,44,36,43)(39,48,42,47);(1,3)(2,7)(4,12)(5,11)(6,13)(8,14)(9,15)(10,16)(17,19)(18,24)(20,33)(21,30)(22,38)(23,31)(25,42)(26,40)(27,41)(28,37)(29,35)(32,39)(34,48)(36,47)(43,46)(44,45);(2,8)(3,11)(4,5)(9,10)(12,13)(14,15)(17,20,22,34,23,35,21,36)(18,25,26,44,27,32,37,43)(19,29,30,48,31,33,38,47)(24,39,28,45,41,42,40,46);(1,4,6,5)(2,8,10,9)(3,12,13,11)(7,14,16,15)(17,21,23,22)(18,26,27,37)(19,30,31,38)(20,34,35,36)(24,40,41,28)(25,43,32,44)(29,47,33,48)(39,45,42,46)

Polycyclic presentation:

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

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, 146038);