#487333

order	512
nilpotency class	4
isoclinism	isoclinic class of #486859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#29679	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

765 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 765 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394529);