#6344209

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	#62833	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2786 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 6251405);