#120902

order	512
nilpotency class	3
isoclinism	isoclinic class of #120900
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 389 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) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fi,e^c=eh,f^c=fh,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, 28098);