#482012

order	512
nilpotency class	4
isoclinism	isoclinic class of #481837
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

853 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18)(19,27,30,42)(20,31)(21,35,37,25)(22,26)(23,29)(24,33)(28,44,32,43)(34,41,48,46)(36,39)(38,47)(40,45);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,30)(18,23,26,39)(20,32,33,28)(21,36,37,29)(24,40,31,38)(25,42,35,27)(34,45,48,47)(41,43,46,44);(2,8)(7,15)(9,10)(11,16)(17,20,22,33)(18,24,26,31)(19,28,30,32)(21,34,37,48)(23,38,39,40)(25,41,35,46)(27,43,42,44)(29,45,36,47);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(20,33)(24,31)(28,32)(34,48)(38,40)(41,46)(43,44)(45,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,d^c=di,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, 389208);