#115128

order	512
nilpotency class	3
isoclinism	isoclinic class of #115124
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#60801	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 871 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257487
maximal normal subgroup of	#72257487
maximal quotient of	#72257698

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22324);