#484121

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	#33733	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

441 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)(17,18,22,27,23,26,38,28)(19,29,32,40,31,46,33,48)(20,25,34,44,35,43,21,36)(24,39,41,47,42,37,45,30);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(17,19)(18,24)(20,30)(21,37)(22,32)(23,31)(25,40)(26,42)(27,41)(28,45)(29,36)(33,38)(34,39)(35,47)(43,48)(44,46);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(17,20,22,34,23,35,38,21)(18,25,27,44,26,43,28,36)(19,30,32,39,31,47,33,37)(24,40,41,46,42,48,45,29);(3,11)(4,13)(7,10)(12,15)(18,26)(19,31)(25,43)(27,28)(30,47)(32,33)(36,44)(37,39)

Polycyclic presentation:

2^9|a^2=f,c^2=f,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.

Relationships

maximal subgroup of	#72260370, #72260372, #72260405
maximal normal subgroup of	#72260370, #72260372, #72260405

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391317);