#485356

order	512
nilpotency class	4
isoclinism	isoclinic class of #485320
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×C₂²⋊C₄	C₂⁴
prime radical	#534	C₂³
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

734 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,29)(20,27)(21,34)(22,25)(23,28)(24,30)(26,37)(31,47)(32,44)(33,41)(35,45)(36,43)(38,40)(39,48)(42,46);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,32)(18,24,28,40)(20,30,22,38)(21,35,37,31)(25,29,27,44)(26,42,34,39)(33,48,36,46)(41,47,43,45);(17,20)(18,25,28,27)(19,30)(21,33)(22,23)(24,29,40,44)(26,41,34,43)(31,46)(32,38)(35,48)(36,37)(39,45,42,47);(2,7)(6,15)(8,12)(9,16)(17,21,23,37)(18,26,28,34)(19,31,32,35)(20,33,22,36)(24,39,40,42)(25,41,27,43)(29,45,44,47)(30,46,38,48)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,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, 392552);