#122779

order	512
nilpotency class	4
isoclinism	isoclinic class of #122694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

322 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,9,11,16)(4,12,14,8)(5,7)(6,10)(13,15)(17,18,21,27,23,29,22,28)(19,30,33,46,35,47,34,44)(20,36,38,26,40,45,39,25)(24,37,41,32,43,48,42,31);(1,3)(2,6)(4,13)(5,11)(7,15)(8,16)(9,12)(10,14)(17,19,23,35)(18,24,29,43)(20,37,40,48)(21,34,22,33)(25,44,26,46)(27,42,28,41)(30,36,47,45)(31,39,32,38);(2,7)(4,14)(9,16)(10,13)(17,20,21,38,23,40,22,39)(18,25,28,45,29,26,27,36)(19,31,33,37,35,32,34,48)(24,30,42,44,43,47,41,46)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=gh,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=dhi,e^d=ehi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29975);