#6343427

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343418
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	C₂²×C₄	C₂⁵
prime radical	#62788	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18)(19,27,30,23)(20,25)(21,26)(22,24)(28,41,44,35)(29,42,46,36)(31,39)(32,37)(33,40)(34,38)(43,48,45,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,35)(26,36)(31,43)(32,44)(33,45)(34,46)(37,41)(38,42)(39,47)(40,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,24)(23,27)(25,37)(26,38)(35,41)(36,42)(39,40)(47,48);(17,20,22,32)(18,25,24,37)(19,28,30,44)(21,33,34,31)(23,35,27,41)(26,40,38,39)(29,45,46,43)(36,48,42,47);(17,21,22,34)(18,26,24,38)(19,29,30,46)(20,31,32,33)(23,36,27,42)(25,39,37,40)(28,43,44,45)(35,47,41,48)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=chi,f^a=fh,f^b=fi,g^b=ghi,e^d=eh

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, 6250623);