#238973

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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 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)(3,10)(4,14)(5,8)(6,9)(7,12)(11,16)(13,15)(17,18,23,27)(19,28,32,40)(20,33,36,25)(21,38,22,26)(24,30,41,31)(29,44,46,42)(34,39,45,48)(35,43,37,47);(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,19)(18,24)(20,34)(21,30)(22,31)(23,32)(25,42)(26,40)(27,41)(28,38)(29,37)(33,44)(35,46)(36,45)(39,43)(47,48);(1,3)(2,7)(4,13)(5,12)(6,11)(8,16)(9,15)(10,14)(17,20,21,35,23,36,22,37)(18,25,26,43,27,33,38,47)(19,29,31,45,32,46,30,34)(24,39,28,42,41,48,40,44);(3,11)(7,15)(10,16)(12,13)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,35,36,37)(24,40,41,28)(25,43,33,47)(29,34,46,45)(39,44,48,42)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=h,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, 146169);