#191272

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

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	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

475 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,e^2=i,f^2=h,g^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, 98468);