#6344264

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343584
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=h,f^2=i,g^2=hi,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251460);