#121965

order	512
nilpotency class	3
isoclinism	isoclinic class of #121964
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

406 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,g^b=gi,d^c=di,e^c=ei,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29161);