#483611

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#466	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

420 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)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,23,27,22,29,39,46,24,28,40,45,21,26,38,44)(19,30,33,35,31,42,47,36,34,43,48,37,32,25,41,20);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,25)(20,27)(21,32)(22,31)(23,33)(24,34)(26,43)(28,42)(29,30)(35,46)(36,45)(37,44)(38,41)(39,47)(40,48);(2,8)(7,15)(9,13)(10,16)(17,20,23,30,22,35,39,42,24,36,40,43,21,37,38,25)(18,19,27,33,29,31,46,47,28,34,45,48,26,32,44,41);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(17,21,24,22)(18,26,28,29)(19,31,34,32)(20,35,36,37)(23,38,40,39)(25,30,42,43)(27,44,45,46)(33,47,48,41)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320227, #72320246, #72320272, …
maximal normal subgroup of	#72320227, #72320246, #72320272, …
maximal quotient of	#72323193

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390807);