#137758

order	512
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44954);