#116138

order	512
nilpotency class	3
isoclinism	isoclinic class of #116135
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

559 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fi,d^c=dh,e^c=eh,f^c=fhi,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, 23334);