#480614

order	512
nilpotency class	4
isoclinism	isoclinic class of #480476
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

856 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387810);