#118605

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

549 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)(3,9)(4,13)(5,8)(6,12)(7,15)(10,14)(11,16)(17,18,22,26,23,27,21,37)(19,28,31,40,32,39,30,24)(20,33,35,25,36,42,34,43)(29,41,46,48,47,38,45,44);(1,2)(3,10,11,6)(4,7)(5,8)(9,12,16,14)(13,15)(17,19,21,30,23,32,22,31)(18,24,26,28,27,40,37,39)(20,29,34,45,36,47,35,46)(25,41,42,48,43,38,33,44);(1,3)(2,6)(4,14)(5,11)(7,16)(8,10)(9,13)(12,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,36)(24,38)(26,42)(27,43)(28,44)(30,45)(31,46)(32,47)(33,37)(39,48)(40,41)

Polycyclic presentation:

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

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, 25801);