#116263

order	512
nilpotency class	3
isoclinism	isoclinic class of #10326
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

610 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)(3,11)(4,7)(5,10)(6,9)(8,13)(12,15)(14,16)(17,18,21,26,22,25,33,39)(19,23,29,35,28,36,41,46)(20,24,32,38,31,37,44,47)(27,34,40,45,30,42,43,48);(2,7)(3,12,14,13)(8,11,15,16)(9,10)(17,19,22,28)(18,23,25,36)(20,30)(21,29,33,41)(24,34)(26,35,39,46)(27,31)(32,43)(37,42)(38,45)(40,44)(47,48);(1,3)(2,8)(4,14)(5,12)(6,13)(7,15)(9,16)(10,11)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,37)(26,38)(28,30)(29,40)(33,44)(35,45)(36,42)(39,47)(41,43)(46,48)

Polycyclic presentation:

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

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, 23459);