#486592

order	512
nilpotency class	4
isoclinism	isoclinic class of #486147
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

738 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285064, #72285101, #72285325
maximal normal subgroup of	#72285064, #72285101, #72285325

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393788);