#484124

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	#2215	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

418 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)(17,18,23,27,24,29,21,28)(19,30,32,40,34,47,33,46)(20,26,35,44,36,45,22,37)(25,39,42,48,43,38,41,31);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(17,19)(18,25)(20,31)(21,33)(22,38)(23,32)(24,34)(26,40)(27,42)(28,41)(29,43)(30,37)(35,39)(36,48)(44,47)(45,46);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(17,20,23,35,24,36,21,22)(18,26,27,44,29,45,28,37)(19,31,32,39,34,48,33,38)(25,40,42,47,43,46,41,30);(3,11)(4,13)(7,10)(12,15)(17,21,24,23)(18,27,29,28)(19,32,34,33)(20,22,36,35)(25,41,43,42)(26,44,45,37)(30,46,47,40)(31,39,48,38)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=i,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, 391320);