#231515

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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

591 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138711);