#480758

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

839 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)(19,29,32,44)(20,25)(21,26)(22,36,37,27)(23,28)(24,31)(30,43,34,47)(33,42,35,40)(38,45)(39,46)(41,48);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,15)(10,14)(17,19)(18,24,28,39)(20,30)(21,34)(22,31)(23,32)(25,38,26,41)(27,29,36,44)(33,45)(35,48)(37,46)(40,43,42,47);(1,4,6,5)(2,8,9,10)(3,11,12,13)(7,14,15,16)(17,20,23,21)(18,25,28,26)(19,30,32,34)(22,33,37,35)(24,38,39,41)(27,40,36,42)(29,43,44,47)(31,45,46,48);(2,9)(7,15)(8,10)(14,16)(17,21,23,20)(18,26,28,25)(19,30,32,34)(22,35,37,33)(24,38,39,41)(27,42,36,40)(29,43,44,47)(31,45,46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=i,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, 387954);