#150049

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	#532	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

258 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,5)(4,7)(6,8)(9,10,12,18,14,19,28,40,15,20,29,41,32,42,54,64,13,17,27,36,30,37,52,59,31,38,53,60,55,61,69,71)(11,21,23,39,24,43,47,62,25,44,48,63,49,65,67,72,22,16,26,33,45,34,50,56,46,35,51,57,66,58,68,70);(2,5)(3,8)(6,7)(9,11)(10,16)(12,26)(13,22)(14,24)(15,25)(17,21)(18,39)(19,34)(20,35)(23,27)(28,50)(29,51)(30,45)(31,46)(32,49)(33,36)(37,43)(38,44)(40,62)(41,63)(42,58)(47,52)(48,53)(54,68)(55,66)(56,59)(57,60)(61,65)(64,72)(67,69)(70,71);(10,17)(11,22)(18,36)(19,37)(20,38)(23,26)(24,45)(25,46)(40,59)(41,60)(42,61)(47,50)(48,51)(49,66)(64,71)(67,68)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57245);