#136067

order	512
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1616
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8857
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

333 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,13,16,39)(4,18)(5,22)(6,27)(7,12)(8,15,31,45)(9,20)(10,26)(11,29)(14,40,42,38)(17,46,28,35)(19,50)(21,34)(23,32,58,37)(24,55)(25,56)(30,44,47,41)(33,59)(36,60)(43,62,54,64)(48,52)(49,53)(51,63,57,61);(1,3,6,28,7,16,29,17)(2,8,11,37,12,31,27,32)(4,14,21,54,20,42,53,43)(5,23,26,15,25,58,60,45)(9,30,34,63,18,47,49,61)(10,35,22,13,36,46,56,39)(19,51,52,41,24,57,59,44)(33,62,55,38,48,64,50,40);(1,4,7,20)(2,9)(3,14,16,42)(5,24,25,19)(6,21,29,53)(8,30)(10,33)(11,34)(12,18)(13,38)(15,44,45,41)(17,43,28,54)(22,55)(23,57,58,51)(26,59,60,52)(27,49)(31,47)(32,61)(35,62)(36,48)(37,63)(39,40)(46,64)(50,56)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=h,d^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281031, #72281057, #72281268
maximal normal subgroup of	#72281031, #72281057, #72281268

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43263);