#485871

order	512
nilpotency class	4
isoclinism	isoclinic class of #485657
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	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 723 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393067);