#484123

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
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	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

401 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391319);