#122775

order	512
nilpotency class	4
isoclinism	isoclinic class of #122694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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

305 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 68 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)(6,10,14,16)(11,15)(17,18,22,28,24,30,23,29)(19,31,34,41,36,46,35,45)(20,37,21,40,39,27,38,26)(25,32,42,33,44,48,43,47);(1,3,5,11)(2,6,8,14)(4,10,13,16)(7,9,12,15)(17,19,24,36)(18,25,30,44)(20,33,39,47)(21,32,38,48)(22,35,23,34)(26,31,40,46)(27,41,37,45)(28,43,29,42);(4,13)(6,14)(7,12)(10,16)(17,20,23,38,24,39,22,21)(18,26,28,37,30,40,29,27)(19,32,35,47,36,48,34,33)(25,41,42,46,44,45,43,31)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=g,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, 29971);