#485752

order	512
nilpotency class	4
isoclinism	isoclinic class of #485649
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 778 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^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,d^c=di,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, 392948);