#251948

order	512
nilpotency class	3
isoclinism	isoclinic class of #251935
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^c=fh,f^d=fh

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, 159144);