#151035

order	512
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×D₁₆
commutator	C₁₆	C₂²×C₈
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂	C₂×#1371
prime radical	C₂³	#368
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

141 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,11,7,12,26,38,8,14,27,39,29,13,6,28)(3,15,19,42,21,43,50,63,17,40,49,62,51,44,20,53)(4,22,23,35,24,54,47,37,25,56,57,36,46,55,16,10)(9,30,32,52,33,59,61,48,31,58,60,64,34,45,41,18);(1,3,16,45,6,20,46,64,29,51,57,58,27,49,25,48,8,17,47,59,26,50,24,52,7,21,23,30,5,19,4,18)(2,9,22,42,11,32,35,43,12,33,54,63,38,61,37,40,14,31,56,62,39,60,36,44,13,34,55,53,28,41,10,15);(3,17)(9,31)(15,40)(18,48)(19,49)(20,50)(21,51)(30,58)(32,60)(33,34)(41,61)(42,62)(43,44)(45,59)(52,64)(53,63)

Polycyclic presentation:

2^9|a^2=e,b^2=df,d^2=fh,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225305, #72225308, #72225314, …
maximal normal subgroup of	#72225305, #72225308, #72225314, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58231);