#482829

order	512
nilpotency class	4
isoclinism	isoclinic class of #482728
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#556 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

745 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,d^c=dh,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, 390025);