#6344251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343554
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	#62838	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2656 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:

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

Polycyclic presentation:

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

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, 6251447);