#138123

order	512
nilpotency class	4
isoclinism	isoclinic class of #137984
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 343 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45319);