#121135

order	512
nilpotency class	3
isoclinism	isoclinic class of #121133
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

351 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28331);