#480755

order	512
nilpotency class	4
isoclinism	isoclinic class of #480684
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

819 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387951);