#117978

order	512
nilpotency class	3
isoclinism	isoclinic class of #10496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
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

537 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:

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

Polycyclic presentation:

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

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, 25174);