#482248

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
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×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

883 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)(19,27,29,40)(20,24,32,39)(21,33,35,25)(22,26)(23,30)(28,41)(31,44)(34,37)(36,45,47,43)(38,46)(42,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(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);(2,8)(7,15)(9,10)(11,16)(17,20)(18,24)(19,28)(21,31)(22,32)(23,36)(25,38)(26,39)(27,41)(29,42)(30,43)(33,44)(34,45)(35,46)(37,47)(40,48);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(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,d^2=hi,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^c=fi,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, 389444);