#112171

order	512
nilpotency class	3
isoclinism	isoclinic class of #112171
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60802	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

879 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 879 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,9,11,15)(4,7)(5,8)(6,10,14,12)(13,16)(17,18)(19,27,29,23)(20,31,33,24)(21,26)(22,25)(28,45)(30,43,35,39)(32,48)(34,44,36,41)(37,42)(38,46)(40,47);(1,3)(2,6)(4,12)(5,11)(7,15)(8,14)(9,16)(10,13)(17,19)(18,23)(20,32)(21,35)(22,29)(24,40)(25,27)(26,43)(28,34)(30,37)(31,48)(33,47)(36,46)(38,41)(39,42)(44,45);(4,13)(6,14)(7,16)(10,12)(17,20)(18,24)(19,28)(21,36)(22,33)(23,38)(25,31)(26,44)(27,45)(29,46)(30,47)(32,35)(34,37)(39,48)(40,43)(41,42)

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh,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, 19367);