#129476

order	512
nilpotency class	4
isoclinism	isoclinic class of #129411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

754 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fg,d^c=di,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36672);