#129128

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#107
prime radical	C₂³	#427
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

238 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 118 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)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,22,28,23,29,21,27,24,30,38,46,37,45,20,26)(19,31,34,25,35,42,39,43,36,47,32,44,40,41,33,48);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,25)(20,32)(21,34)(22,39)(23,40)(24,36)(26,41)(27,42)(28,47)(29,48)(30,44)(31,46)(33,38)(35,37)(43,45);(2,7)(6,12)(8,14)(13,16)(17,20,37,38,24,21,23,22)(18,26,45,46,30,27,29,28)(19,32,35,33,36,34,40,39)(25,41,43,31,44,42,48,47)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ch,d^b=dg,e^b=eg,g^b=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36324);