#482479

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
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	#35210	C₂
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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=i,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,f^d=fi,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, 389675);