#150053

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#102
commutator	C₈	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂	C₂×D₄⋊C₁₆
prime radical	C₂×#135	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

249 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=f,e^2=g,f^2=i,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fi,c^b=ci,d^b=dfi,e^b=ei,f^b=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225516, #72225517, #72225518
maximal normal subgroup of	#72225516, #72225517, #72225518

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57249);