#138382

order	512
nilpotency class	4
isoclinism	isoclinic class of #138195
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

413 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,11,12,15,14,16,13,7)(17,18,20,25,35,45,47,39,22,27,36,38,21,26,33,23)(19,28,29,37,30,43,46,44,32,42,34,48,31,24,40,41);(1,3)(2,7,10,15)(4,13)(5,12)(6,14)(8,16,9,11)(18,23)(20,33)(21,35)(24,37)(25,26)(27,39)(28,41)(29,40)(30,31)(34,46)(36,47)(38,45)(42,44)(43,48);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,15,16)(17,19)(18,24)(20,34)(21,31)(22,32)(23,37)(25,44)(26,42)(27,43)(28,45)(29,36)(30,35)(33,46)(38,41)(39,48)(40,47)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=ghi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=ei,f^c=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, 45578);