#238977

order	512
nilpotency class	3
isoclinism	isoclinic class of #238970
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	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

788 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)(17,18,23,26)(19,28,32,43)(20,33,36,25)(21,38,22,27)(24,31,40,30)(29,44,46,41)(34,39,45,48)(35,42,37,47);(1,3)(2,7)(4,12)(5,11)(6,13)(8,14)(9,15)(10,16)(17,19,23,32)(18,24,26,40)(20,34,36,45)(21,31,22,30)(25,41,33,44)(27,43,38,28)(29,35,46,37)(39,47,48,42);(2,8)(3,11)(4,5)(9,10)(12,13)(14,15)(17,20,21,35,23,36,22,37)(18,25,27,42,26,33,38,47)(19,29,30,45,32,46,31,34)(24,39,28,41,40,48,43,44);(1,4,6,5)(2,8,10,9)(3,12,13,11)(7,14,16,15)(18,26)(24,40)(25,33)(27,38)(28,43)(39,48)(41,44)(42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146173);