#483762

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
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 ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

414 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320209, #72320219, #72320272, …
maximal normal subgroup of	#72320209, #72320219, #72320272, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390958);