#191251

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

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,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, 98447);