#481023

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
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×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#556 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

797 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=h,d^2=hi,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,f^b=fgi,d^c=dh,f^e=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388219);