#483767

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

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,c^a=ci,d^a=dg,e^a=eg,d^b=di,e^b=eg,d^c=dgi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286931, #72286988, #72287023, …
maximal normal subgroup of	#72286931, #72286988, #72287023, …
maximal quotient of	#72323196

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390963);