#6344295

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343678
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	#62833	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2749 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^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,e^c=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251491);