#6344365

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₄⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1571 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=h,f^2=h,g^2=i,b^a=bf,d^a=di,f^a=fh,c^b=ci,e^c=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251561);