#116253

order	512
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
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

564 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fi,d^c=dh,e^c=ehi,f^c=fhi,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, 23449);