#137267

order	512
nilpotency class	4
isoclinism	isoclinic class of #137153
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

335 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=g,e^2=i,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ehi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44463);