#110953

order	512
nilpotency class	3
isoclinism	isoclinic class of #110922
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

631 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18,21,28,23,26,22,27)(19,24,31,40,30,41,37,39)(20,32,35,25,34,43,36,44)(29,45,46,38,33,47,48,42);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19,22,31,23,30,21,37)(18,24,28,39,26,41,27,40)(20,33,36,48,34,29,35,46)(25,42,43,47,44,38,32,45);(3,10)(6,14)(9,15)(11,16)(17,20,23,34)(18,25)(19,29,30,33)(21,35,22,36)(24,38)(26,44)(27,32)(28,43)(31,46,37,48)(39,45)(40,47)(41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18149);