#482934

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×C₂²⋊C₄	C₂⁴
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#553 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

816 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18)(19,27)(20,24)(21,34,36,25)(22,26)(23,29,38,35)(28,43)(30,42)(31,45)(32,47,48,40)(33,41)(37,44,39,46);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19,22,30)(18,23)(20,31,33,28)(21,35,36,29)(24,39)(25,42)(26,38)(27,34)(32,44,48,46)(37,41)(40,43)(45,47);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,20,22,33)(18,24,26,41)(19,28,30,31)(21,32,36,48)(23,37,38,39)(25,40,34,47)(27,43,42,45)(29,44,35,46);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=h,d^2=hi,e^2=hi,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fghi,d^c=dhi,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, 390130);