#136834

order	512
nilpotency class	4
isoclinism	isoclinic class of #2204
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

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

normal subgroups only

This group has 493 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:

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

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,c^b=ci,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

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343179, #72343182, #72343189
maximal normal subgroup of	#72343179, #72343182, #72343189

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44030);