#483950

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#466	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=dh,e^a=eg,g^a=gi,d^b=dh,e^b=eg,g^b=gi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391146);