#136833

order	512
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8965
prime radical	#3018	C₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=egi,g^a=gi,d^b=di,e^b=egi,f^b=fi,d^c=dg,e^c=ei,f^c=fg,e^d=ei,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343161, #72343179, #72343187
maximal normal subgroup of	#72343161, #72343179, #72343187

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44029);