#121921

order	512
nilpotency class	3
isoclinism	isoclinic class of #121919
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	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

386 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,11,14,15)(4,7)(5,9)(6,10)(8,13,16,12)(17,18,22,26,23,27,21,37)(19,28,31,40,30,24,38,41)(20,32,35,44,36,43,34,25)(29,42,47,45,46,48,33,39);(2,7,10,9)(8,15,16,11)(17,19,21,38,23,30,22,31)(18,24,26,41,27,28,37,40)(20,33,34,46,36,47,35,29)(25,42,32,45,44,48,43,39);(1,3)(2,8,10,16)(4,12)(5,13)(6,14)(7,15,9,11)(17,20,21,34,23,36,22,35)(18,25,26,32,27,44,37,43)(19,29,38,33,30,46,31,47)(24,39,41,42,28,45,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29117);