#121199

order	512
nilpotency class	3
isoclinism	isoclinic class of #121197
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

360 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 108 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)(17,18,22,28,23,26,21,27)(19,24,32,40,30,41,31,39)(20,33,35,43,36,44,37,25)(29,45,34,48,46,42,47,38);(2,7,10,9)(8,15,16,11)(17,19,21,32,23,30,22,31)(18,24,28,39,26,41,27,40)(20,34,35,29,36,47,37,46)(25,42,44,38,43,45,33,48);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)(17,20)(18,25,26,43)(19,29,30,46)(21,35)(22,37)(23,36)(24,38)(27,33,28,44)(31,34,32,47)(39,45)(40,42)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,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, 28395);