#483674

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
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	#447	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320272, #72322030, #72322049
maximal normal subgroup of	#72320272, #72322030, #72322049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390870);