#122753

order	512
nilpotency class	4
isoclinism	isoclinic class of #122704
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊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×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

307 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29949);