#118276

order	512
nilpotency class	3
isoclinism	isoclinic class of #9974
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

555 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:

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

Polycyclic presentation:

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

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, 25472);