#6344072

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62815	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2835 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251268);