#263393

order	512
nilpotency class	3
isoclinism	isoclinic class of #263387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62365	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

825 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:

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

Polycyclic presentation:

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

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, 170589);