#120925

order	512
nilpotency class	3
isoclinism	isoclinic class of #120924
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	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 375 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,g^b=ghi,e^c=ehi,f^c=fh,g^c=gi

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, 28121);