#482216

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60707	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di

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, 389412);