#481881

order	512
nilpotency class	4
isoclinism	isoclinic class of #481837
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 ◁ C₂²×D₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

864 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,21,26,22,25,33,39)(19,23,29,35,28,36,44,40)(20,30,32,45,31,24,37,38)(27,41,43,48,42,34,46,47);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,23,25,36)(20,27)(21,29)(22,28)(24,34,30,41)(26,40,39,35)(31,42)(32,43)(33,44)(37,46)(38,48,45,47);(2,7)(6,14)(8,15)(9,16)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,30)(26,38)(28,42)(29,43)(33,37)(35,47)(36,41)(39,45)(40,48)(44,46);(20,31)(24,30)(27,42)(32,37)(34,41)(38,45)(43,46)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,d^c=di,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, 389077);