#122412

order	512
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
socle	C₂²	#1459
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

294 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,9)(4,12,13,7)(5,8)(6,11,14,16)(10,15)(17,18,22,28,23,29,36,43,24,30,20,26,21,27,37,44)(19,31,34,47,33,42,48,40,35,45,32,46,38,25,39,41);(1,3,5,10)(2,6,8,14)(4,11,13,16)(7,9,12,15)(17,19)(18,25)(20,34)(21,38)(22,32)(23,33)(24,35)(26,41)(27,45)(28,40)(29,31)(30,42)(36,39)(37,48)(43,46)(44,47);(3,10)(6,14)(9,15)(11,16)(17,20,23,37,24,22,21,36)(18,26,29,44,30,28,27,43)(19,32,33,39,35,34,38,48)(25,40,31,46,42,41,45,47)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=g,d^2=i,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=chi,d^b=di,e^b=efi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29608);