#486692

order	512
nilpotency class	4
isoclinism	isoclinic class of #486689
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18203
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

663 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 663 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gi,e^2=g,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=egi,f^b=fhi,g^b=gi,e^c=eh,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281586, #72281689, #72281886, …
maximal normal subgroup of	#72281586, #72281689, #72281886, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393888);