#231507

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	C₂³	#518
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

615 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)(19,30,34,45)(20,35)(21,26)(22,27)(23,29)(24,32,42,33)(25,37)(28,36)(31,44,47,40)(38,43)(39,48,46,41);(1,3,5,11,6,2,7,9)(4,10,13,16,12,8,14,15)(17,19,22,33,23,34,36,32)(18,24,27,45,29,42,28,30)(20,31,37,46,38,47,21,39)(25,40,43,48,26,44,35,41);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,20,36,21,23,38,22,37)(18,25,27,43,29,26,28,35)(19,31,32,39,34,47,33,46)(24,40,45,48,42,44,30,41);(4,12)(8,10)(13,14)(15,16)(17,21,36,38,23,37,22,20)(18,26,28,43,29,25,27,35)(19,31,33,46,34,47,32,39)(24,41,45,40,42,48,30,44)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=eh,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, 138703);