#483765

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	C₂×#135	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,7,12)(3,13,4,9)(5,11,6,10)(8,15,16,14)(17,18,21,24,22,25,29,31,20,19,23,26,28,27,30,32);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,10,13,11)(18,19)(24,26)(25,27)(31,32);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,10,16,11)(17,19,21,26,22,27,29,32,20,18,23,24,28,25,30,31);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,16,9)(18,19)(24,26)(25,27)(31,32)

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,c^b=ci,e^b=eg,d^c=dgi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287023, #72287103, #72287116, …
maximal normal subgroup of	#72287023, #72287103, #72287116, …
maximal quotient of	#72323196

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390961);