#280932

order	512
nilpotency class	3
isoclinism	isoclinic class of #280915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 451 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=eh,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

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, 188128);