#482502

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
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×C₄⋊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×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

845 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18,21,26,22,25,34,41)(19,27,29,42,30,23,35,37)(20,24,32,40,33,39,45,48)(28,38,43,47,31,36,44,46);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,30)(18,23,25,27)(20,31,33,28)(21,29,34,35)(24,38,39,36)(26,42,41,37)(32,44,45,43)(40,46,48,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,33)(18,24,25,39)(19,28,30,31)(21,32,34,45)(23,36,27,38)(26,40,41,48)(29,43,35,44)(37,46,42,47);(18,25)(21,34)(23,27)(24,39)(29,35)(32,45)(36,38)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,e^2=h,f^2=i,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, 389698);