#191250

order	512
nilpotency class	3
isoclinism	isoclinic class of #191246
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

534 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)(3,9)(4,14)(5,10)(6,15)(7,11)(8,12)(13,16)(17,18)(19,29,33,24)(20,34,36,42)(21,26)(22,28)(23,27)(25,35,41,37)(30,44)(31,40,32,39)(38,45)(43,47)(46,48);(1,3,6,12,7,13,5,4)(2,8,10,9,11,14,15,16)(17,19,22,32,23,33,21,31)(18,24,28,40,27,29,26,39)(20,30,35,45,36,46,37,47)(25,38,34,48,41,43,42,44);(1,4,5,13,7,12,6,3)(2,9,15,8,11,16,10,14)(17,20,23,36)(18,25)(19,30,33,46)(21,37,22,35)(24,38)(26,42)(27,41)(28,34)(29,43)(31,47,32,45)(39,44)(40,48);(1,5,7,6)(2,10,11,15)(3,12,13,4)(8,16,14,9)(17,21,23,22)(18,26,27,28)(19,31,33,32)(20,35,36,37)(24,39,29,40)(25,34,41,42)(30,45,46,47)(38,48,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98446);