#231505

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

641 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138701);