#102054

order	512
nilpotency class	3
isoclinism	isoclinic class of #102045
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

438 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cgi,d^b=dh,f^b=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9250);