#119377

order	512
nilpotency class	3
isoclinism	isoclinic class of #119374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

699 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26573);