#119577

order	512
nilpotency class	3
isoclinism	isoclinic class of #119572
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	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

524 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,e^c=eh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26773);